-fA\ '<>t' .- .V^-H^^^Vv .-^if^- , ■^t^- # ^^fe^^--|^''^ .-?' /• ^,;V,.^->' ^^ ^-A. ['■^ :''■'■ ' :4*^ ^. Digitized by tine Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarytreatiOOricerich T^c 4^^ ff^ AN ELEMENTARY TREATISE ON THE DIFFERENTIAL CALCULUS FOUNDKD ON THE METHOD OF RATES OR FLUXIONS BY JOHN MINOT RICE PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY AND WILLIAM WOOLSEY JOHNSON FKOFESSOR OF MATHEMATICS IN SAINT JOHN's COLLEGE ANNAPOLIS MARYLAND ABRIDGED EDITION THIRD THOUSAND. NEW YORK: JOHN WILEY AND SONS, .53 East Tenth Street, 1S93. ^^L.J~iA^ X^jt^ Copyright, 1880, John Wiley and Sons. ^^C/t^f^--^^ /';^ Now York : J. J. Uttle & Co., Prlntew, 10 to 30 Astor Flace. PREFACE In preparing this abridgment of their treatise on the Dif- ferential Calculus, the authors have endeavored to adapt it to the wants of those instructors who find the larger work too extensive for the time allotted to this subject. J. M. R. W. W. J. Annapolis, Maryland, Augtisif 1880. ivi57?C ^ / CONTENTS. CHAPTER I. r Functions, Rates, and Derivatives. I. PAGE Functions I Implicit functions 3 Inverse functions 4 Classification of functions 4 Expressions involving an unknown function 5 Examples 1 6 II. Rates 9 Constant rates 10 Variable velocities ii Illustration by means of Attwood's machine 11 The measure of a variable rate 12 Differentials 12 The differentials of polynomials 13 The differential of vix 14 Examples II 15 III. The differentials of functions 16 The derivative — its value independent of dx 17 The geometrical meaning of the derivative 19 Examples III 21 CHAPTER II. The Differentiation of Algebraic Functions. IV. The square 23 The square root 25 Examples IV 26 CONTENTS. V. PAGE The product 29 The reciprocal 30 The quotient 31 The power ~32~ Examples V 34 CHAPTER III. The Differentiation of Transcendental Functions. VI. The logarithm 37 The Napierian base 39 The logarithmic curve (jv — log^jr) 40 Logarithmic differentiation 41 Differentials of algebraic functions deduced by logarithmic differentiation 42 Exponential functions 43 Examples VI 44 VII. Trigonometric or circular functions ; 47 The sine and the cosine 48 The tangent ar^d the cotangent 49 The secant and the cosecant 50 The versed sine 50 Examples VII 51 VIII. Inverse circular functions— their primary values 54 The inverse sine and the inverse cosine 56 The inverse tangent and the inverse cotangent 57 The inverse secant and the inverse cosecant 58 The inverse versed-sine 59 Examples involving trigonometric reductions 59 Examples VIII 60 IX. Differentials of functions of two variables 62 Examples IX 64 Miscellaneous examples of differentiation 65 vi CONTENTS. CHAPTER IV. Successive Differentiation. Velocity and acceleration 67 Component velocities and accelerations 69 Examples X : 70 XI. Successive derivatives 73 The geometrical meaning of the second derivative 73 Points of inflexion 74 Successive differentials • 75 Equicrescent variables • • 75 Examples XI • • . . - 76 CHAPTER V. The Evaluation of Indeterminate Forms. XII. Indeterminate or illusory forms 79 Evaluation by differentiation 80 Examples involving decomposition 82 Examples XII 84 XIII. The form ^ 87 Derivatives of functions which assume an infinite value 89 The form o 00 . . . . : 00 The form 00 — 00 no Examples XIII ^. gi XIV. Functions whose logarithms take the form o -oo 93 The form i* g3 The form 0° 04 Examples XIV. g5 CONTENTS. VU CHAPTER VI. Maxima and Minima of Functions of a Single Variable. XV. Conditions indicating the existence of maxima and minima 97 Maxima and minima of geometrical magnitudes 99 Examples XV • loi XVI. Method of discriminating between maxima and minima 103 Alternate maxima and minima 104 The employment of a substituted function 106 Examples XVI 107 XVII. Employment of derivatives higher than the first 109 Complete criterion for a maximum or a minimum ill Infinite values of the derivative 113 Examples XVII 114 Miscellaiuous examples of maxima and minima 115 CHAPTER VII. The Development of Functions in Series. XVIII. The nature of an infinite series I19 Convergent and divergent series 121 Taylor's theorem 122 Lagrange's expression for the remainder 124 The binomial theorem 126 Examples XVIII 127 XIX. Maclaurin's theorem 129 The exponential series and the value of /? 129 Logarithmic series 131 Computation of Napierian logarithms 132 The modulus of tabular logarithms 134 The developments of sin x and of cos x 134 Examples XIX 135 Vlll CONTENTS. CHAPTER VIII. Curve Tracing. XX. PAGE Equations in the form y — f{x) '. 138 Asymptotes parallel to the coordinate axes 138 Minimum ordinates and points of inflexion 140 Oblique asymptotes 141 Curvilinear asymptotes 143 Examples XX 144 XXI. Curves given by polar equations 146 Asymptotes determined by means of polar equations 148 Asymptotic circles 149 Examples XXI 150 XXII. The parabola of the nih. degree ... 151 The cubical and the semicubical parabolas 152 The cissoid of Diodes 153 The cardioid 154 The lemniscata of Bernoulli 154 The logarithmic or equiangular spiral 155 The loxodromic curve 155 The cycloid 157 The epicycloid 139 The hypocycloid 160 The four-cusped hypocycloid 161 CHAPTER IX. Applications of the Differential Calculus to Plane Curves. XXIII. The equation of the tangent 162 The equation of the normal 163 Subtangents and subnormals , 164 The perpendicular from the origin upon a tangent 165 Examples XXIII i66 CONTENTS. IX XXIV. PAGE Polar coordinates 167 Polar subtangents and subnormals 169 The perpendicular from the pole upon a tangent 170 The perpendicular upon an asymptote I7I_ Points of inflexion 171 Exaviples XXIV 173 XXV. Curv'ature 174 The direction of the radius of curvature 176 The radius of curvature in rectangular coordinates 177 Expressions for p in which x is not the independent variable 178 Examples XXV 179 XXVI. Envelopes 180 Two variable parameters 183 Evolutes 185 Examples XXVI 188 CHAPTER X. Functions of Two or More Variables. XXVII. The derivative regarded as the limit of a ratio 190 Partial derivatives 191 Examples XXVII 194 XXVIII. The second derivative regarded as a limit 194 Higher partial derivatives ' 196 Examples XXVIII 198 THE DIFFERENTIAL CALCULUS CHAPTER I. Functions, Rates, and Derivatives. I. Functions, LA quantity whicn depends for its value upon another quantity is said to be 2i function of the latter quantity. Thus x", tan;r, \o^(a 4- x), and a"" are functions of x. The quantity upon which the function depends must be regarded as variable, and be represented in the analytical expression for the function by an algebraic symbol. This quantity is called the independent variable. It is essential that variation of the independent variable should actually produce variation of the function. Thus the quantities ;tr'', x"^ ^r{a ■\- x) {a — x), and (tan x + cot ;ir) sin 2x are not func- tions of X, since each admits of expression in a form which does not involve x, 2. The notation /{x) is employed to denote any function of X, and, when several functions of x occur in the same in- 2 FUNCTIONS RA TES AND DERIVA TIVES. [Art. 2. vestigation, such expressions as F{x\ F' {x), (^ (;r), etc., are employed, the enclosed letter always denoting- the indepen- dent variable. When expressions like f{}),f{a\ f{2x), or /(o) are employed, it must be understood that the enclosed quantity is to be substituted for x in the expression which defines f(x). Thus, if we have f{x) = x' + X, /(i) =z 2, /{2x) = 4x' + 2x, and /(o) = c. Again, if F{x) = log„;t: {a> \) F{i) = o, F{6) = — 00, and F{a) = i. 3. When x denotes the independent variable upon which a function depends, any quantity independent of x is, in con- tradistinction, called a constant ; both when it is an absolute constant, like i, |/2, or tt, and when it is denoted by a symbol, like a, ?/, or y^ to which any value can be assigned. Thus, when a' is denoted by /(^), it is considered simply as a func- tion of X, and a is regarded as a constant. When it is desired to express that a quantity is a function of two quantities, both the symbols denoting them are placed between marks of parenthesis. Thus, since a^ is a function of X and a, we may write fixy d) = a^. Accordingly we have Ay.b) = b\ /(3, 2) = 8, and /(2, 3) = 9- • 4. It is often convenient to represent the value of a func- tion of X by a single letter ; thus, for example, y = x^. When this notation is used, if we represent the independent variable X by the abscissa of a point, and the function y by the corre- § I.] IMPLICIT FUNCTIONS, 3 spending ordinate, a curve may be constructed which will graphically represent the function, and will serve to illustrate its peculiarities. Rectangular coordinates are usually employed for this purpose. See diagram, Art. lo. A function of the form y z=^ mx -\- b^ m and b being constants, is represented by a straight line. Functions of this form are, for this reason, called linear func- tions, l77tplicit Functions. 5. When an equation is given involving two variables x and J/, either variable is obviously a function of the other ; and the former variable, when its value is not directly ex- pressed in terms of the other, is said to be an implicit func- tion of the latter. Thus, if we have ax"^ — "^axy + y' — ^' = o, either variable is an implicit function of the other. By solving the above equation for x, we obtain /(--f-fl- ,=f±/(.'.f In this form of the equation, x is said to be an explicit func- tion of y. This example will serve to illustrate the fact, that from a single equation involving two variables, there may be derived two or more explicit functions of the same variable. In the above case, x is said to be a two-valued function of y ; while, since the equation is of the third degree in y, the latter is a three-valued function of x. FUNCTIONS RA TES AND DERIVA TIVES. [Art. 6. Inverse Functions. 6. \i y = f{^)y ■*■ is some function oi y ; we may therefore write y—f{x), whence x ±^ (t>{y). Each of the functions /and is then said to be the inverse function of the other. Thus, if y = a"^, we have x = logay ; hence each of these functions is the inverse of the other. So also the square and the square root are inverse functions. 7. In the case of the trigonometric functions, a peculiar notation for the inverse functions has been adopted. Thus, if we have X = sin 6, we write 6 = sin ~^x. Whenever trigonometric functions are employed in the Calculus, the symbol representing the angle always denotes the circu/ar measure of the angle ; that is, the ratio of the arc to the radius. Hence sin~*;ir maybe read either "the in- verse sine of x'' or " the arc whose sine is ;r." The inverse trigonometric functions are evidently many- valued. See Art. 54. The Classification of Functions, 8. With reference to its formy an explicit function is either algebraic or transcendental. An algebraic function is expressed by a definite combination of algebraic symbols, in which the exponents do not involve the independent variable. § I.] THE CLASSIFICATION OF FUNCTIONS. 5 All functions not algebraic are classed as transcendental. Under this head are included exponential functions ; that is, those in which one or more exponents are functions of the variable, as, for example, a^, xa^^, etc. : logarithmic func- tions : the direct and inverse trigonometric functions, and other forms which arise in the higher branches of mathematics. 9. With reference to its mode of variation, a function is said to be an increasing fimction when it increases and de- creases with X ; and a decreasing function when it decreases as X increases, and increases as x decreases. Thus, it is evi- dent that x^ is always an increasing function of x, while — is always a decreasing function of x. Again, tan x is always an increasing function, but sin;ir is sometimes an increasing and sometimes a decreasing function of x, 10. The increase and decrease here considered are algc- hraic. For example, x"^ is an increasing function when x is positive, but when x is negative it becomes a decreasing function ; for, when x is negative and algebraically increasing, x"^ is decreasing. The curve y ^= x"^ which illustrates this function is constructed in Fig. i. Since alge- braic increase in the value of x is represented by motion from left to right, whether the moving point is on the left or on the right of the axis of y, the downward slope of the curve on the left of the origin indicates that x"" is a decreasing function when x is negative. Expressions involving an Unknown Function. 11. An expression involving f{x\ as, for example, xf{x) or F\^f{x)\, is generally a function of x\ but it may happen 6 FUNCTIONS RA TES AND DERIVA TIVES. [Art. 1 1. that such an expression has a value independent of x. Thus, suppose that, in the course of an investigation, the following equation presents itself : — xf{x) = zf{z\ in which/ denotes an unknown function, and x and z are en- tirely independent arbitrary quantities. When this is the case, we can make z a fixed quantity, and give to x any value what- ever; that is, we can make x a variable and z a constant; but if z is a constant, zf(z) is likewise a constant, v/e can, therefore, write xf{x) — c, c being an unknown constant. Hence we have The value of the constant c is readily found, if we know the value of f{x) corresponding to any one value of x. Examples I. 1. {pc) For what value of n does x^ cease to be a function of x} (0) For what values of x does it cease to be a function of n? (a) When n = o. (/3) When ^ = i, or ;ir = o. 2. Ifyfi '- — -^j=,r+ — ^—, show that V is a function of «, but -^ \ a + xj a + X not of X. 3. Show that sin,r tan ^x + cos;r is not a function of x. 4. U y = X + 4/(1 + X-), show that/" — 2xjy is not a function of x, 5. If /(^) = x\ find the value o( /{x + Ji)\ of/(2;r); of /(^'') ; of f{x^-x)', of/(i);/(i2);/[/(^)]. fix + y^) = ;ir' + 2/^ ;r + h^. § I.] EXAMPLES OF FUNCTIONS. 7 6. If /(^) = COS0, find the value of /(o) ; of /(^tt) ; o\ /(i^r) ; of 7. If F{x) — ax, give the value of F{d)\ of F{\)\ of /^(o). Also show that in this case \jF{x)Y — F {7.x). 8. Given j^ — 7.ay + ;ir- = o, make_y an explicit function of x. y — a± ^{0" — X-). 9. Given i + loga _y = 2 log^ {x + «), make ^y an explicit function of x. {x-^ay 10. Given the equations — 71 -\- \ —71 (cos^^' + cos Q' cos + cos'^^, and n — I — n (sin'^^' + sin 6' sin 6 + sin^^) ; eliminate n, and make ^'' an explicit function of 6. Also make n an ex- plicit function of ^. , i e' =zd±^Tz, and ;2 = T sin <^ cosO 11. Given sin — ' jr + sin'" ' y = a, makej/ an explicit function of x. y = sin a 4/(1 — x^) — x cos or. 12. Given tan-';ir + tan~'/ = or /(^)=^/(-^), /(H=,^/«. 21. Given, the property of the same function proved in Example 20; f{mx) = mf{x)\ § I.] EXAMPLES OE EUNCTIONS. by putting 2 for mx, show that and thence deduce the form of the function. See Art. 11. /(^) = ex. 22. Given, [^ {x)Y = [?> (2')> , and 9 (i) = s, ' determine <}> {x). 23. Given (^) + ^ ( j) = {xy) prove ^ {xf") = m (x), and thence prove (p (x) = c logo:. Use the 7)iethods of Examples 19, 20, and 21. 9(^) = e^ II. Rates, 12. In the Differential Calculus, variable quantities are regarded as undergoing continuous variation in magnitude, and the rates of variation, denoted by appropriate symbols, are employed in connection with the values of the variables themselves. If a varying quantity be represented by the distance of a point moving in a straight line from a fixed origin taken on that line, the velocity of the moving point will represent the rate of increase or decrease of the varying quantity. Fig. 2. Thus O (Fig. 2) being the fixed origin and (9/* a variable denoted by x, P is the moving point whose velocity repre- sents the rate of x. The velocity of P, or the rate of x, is regarded as positive when P moves in the direction in which X increases algebraically ; thus, taking the direction OX^ or toward the right, as the positive direction in laying off x, the lO FUNCTIONS RATES AND DERIVATIVES. [Art. 12. velocity is positive when P moves toward the right, whether its position be on the right or on the left of the origin. Ac- cordingly, a rate of algebraic decrease is considered as nega- tive, and would be represented by a point moving toward the left. Constant Rates, (3. The rate of a quantity like the velocity of a point may be either constant or variable. A velocity is uniform or con- stant, when the spaces passed over in any equal intervals of time are equal, or, in other words, wJie7i the spaces passed oveis* in any intervals of time are proportiojial to the intervals. The numerical measure of a uniform velocity is the space passed over in a unit of time ; then if t denote the time elapsed from an assumed origin of time, and k the space passed over by a moving point in a unit of time, kt will denote the space passed over in the time /. Hence, whenever the velocity is uniform, the quotient obtained by dividing the number of units of space by the number of units of time occupied in describing this space is constant, and serves as the numerical measure of the velocity. 14. Now, if ;ir be a quantity having a uniform rate k, it will be represented by the distance from the origin of a point having the uniform velocity k, and if a denote the value of x when / is zero, we shall have X =^ a + kt (i) This formula expresses a uniformly varying quantity as a function of /. When ;r is a uniformly decreasing quantity, k is, of course, negative. Conversely, if x, when expressed as a function of /, is of the form (i), involving the first power only of /, then ;r is a quan- tity having a uniform rate, and the coefficient ^ is a measure of this rate. § n.] VARIABLE VELOCITIES. II Variable Velocities, 15. If the velocity of a point be not uniform, its numerical measure at any instant is the number of units of space which would be described in a unit of time^ were the velocity to remain constant from and after the given instaiit. Thus, when we speak of a body as having at a given in- stant a velocity of 32 feet per second, we mean that should the body continue to move during the whole of the next second, with the same velocity which it had at the given instant, 32 feet would be described. The actual space described may be greater or less, in consequence of the change in velocity which takes place during the second ; it is, for instance, greater than the measure of the velocity at the beginning of the second, in the case of a falling body, because the velocity increases throughout the second. 16. Attwood's machine for determining experimentally the velocities acquired by falling bodies furnishes a familiar example of the practical application of the principle em- bodied in the above definition. This apparatus consists essentially of a thread passing over a fixed pulley, and sustaining equal weights at each ex- tremity, the pulley being so constructed as to offer but slight resistance to turning. On one of the weights a small bar of metal is placed, which, destroying the equilibrium, causes the weight to descend with an increasing velocity. To deter- mine the value of this velocity at any point, a ring is so placed as to intercept the bar at that point, and allow the weight to pass. Thus, the sole cause of the variation of the velocity having been removed, the weight moves on uniformly with the required velocity, and the space described during the next second becomes the measure of this velocity. 12 INUNCTION'S RA TES AND DERIVA TIVES. [Art. I/. Variable Rates. 17. When ;r is a function of /, but not of the form ex- pressed by equation (i), Art. 14— that is, when the function is not linear — the rate of x will be variable.' To obtain the measure of this rate at any given instant, we employ the same principle as in the case of a variable velocity. Thus, let X be represented by O P\s> in Fig. 2, Art. 12, let the sym- bol dt denote an assumed interval of time, and let dx denote the space which would be described in the time dt, were P to move with the velocity which it has at the given instant unchanged throughout the interval of time dt. Then the space which would be described in a unit of time is, evidently, dx . 'dt' which is therefore the measure of the velocity of /*, or the rate of x. This ratio is in general variable, but, when x is of the form a-\- kt/\\, has been shown in Art. 14 that k is the measure of the rate ; we therefore have = k, when X = a -{- kt. dt Differ en tials. (8. The quantities dx and dt are called respectively "the differential of ;r" and "the differential of /." In accordance with the definition of dx given in the pre- ceding article, the differential of a variable quantity at any instant is the increment which would be received in the time dt, were the quantity to continue to increase uniformly during that interval of time with the rate it has at the given § II.] DIFFERENTIALS. 13 instant. The quotient obtained by dividing the differential of any quantity by dt is tJierefore the measure of the rate of the quantity. The differential of a quantity is denoted by prefixing d to the symbol denoting the quantity ; when the symbol denot- ing the quantity is not a single letter it is usually enclosed by marks of parenthesis to avoid ambiguity. Thus, d{x'')y d(xy), ^(tan;r), d{a^ + x^), etc. T/ie Differentials of Polynomials, 19. Let X and y denote two variable quantities, and let a and b denote particular simultaneous values of x and y^ while k and k' denote corresponding values of the rates of x and y. Now, if X and y should continue to vary with these rates, their values would (see Art. 14) be expressed by x = a^kt, and y=b + k't, whence x+y = a-\-b+{k^ k')t. Thus the quantity ;r+ J would become a uniformly varying quantity, and, by Art. 14, its rate would be k-\-k\ which, therefore, is the measure of the rate of x -\-y at the instant when X and y have the rates k and k\ Consequently, dt -^+^ - dt^ dt' Now, since k and k' denote any values of the rates, this equa- tion is universally triie. We have, therefore, d{x^y) = dx^dy (l) This formula is easily extended to the sum of any number of variables. Thus, dix^y \z^r' ")^ dx\d{^y\2-\- * * ^)=^ dx-^-dy-irds-^^ (2) 14 FUNCTIONS RA TES AND DERIVA TIVES. [Art. 20. 20. The differential of a constant is evidently zero, hence d{x-\-h) = dx (3) Again, if y z= — x, y-\-x = Oy hence, by equation (i), since zero is a constant, we have dy -\- dx =^ o, or dy =^ — dx ; that is, d,{—x) = — dx (4) The differential of a negative term is therefore the negative of the differential of the term taken positively. It appears, on combining the results expressed in equations (2), (3), and (4), that t/ie differential of a polynomial is the alge- braic sum of the differentials of its terms; and that constant terms disappear from the result. The Differential of a Term having a Constant Coefficient. 21. Let the term be denoted by mxy m denoting a con- stant. Resuming equation (2), Art. 19 ; viz., d{x -vy^ z^- • • •) = dx ■\- dy -^ dz -\- • . •, and denoting the number of terms by /, we put x=y—z— , thus obtaining d{px)=pdx, (i) p denoting an integer. § II.] THE DIFFERENTIAL OF^mx). 1 5 To extend equation (i) to the case fraction, let in which m denotes a s = - X, then qz=px. By applying equation (i) we obtain qd2=pdxj or d. = ^dx; that is, ^[-A = ~^^' Hence generally, when m is positive, d{in x) — in dx (2) Since d{— x) = — dx, this equation is true likewise when m is negative. It therefore follows that t/ie differential of a term having a constant coefficient is equal to the product of the differential of the variable factor by the constant coefficient. Examples XL I. Find the differential of — , and of 3«' w — 2 * 2dx , dx , and 3<3: m — 2 2. Find the differential of 1^- , and of ^- dx , dx — T, and 2' ^ , ,.«• . , , a + d + (a — b)x dx 3. Find the differential of 7, -r —ri' 4. Find the differential of —7, and of —, — -^ . ^ a -{■ a{a + o) dx J b{dx + dy) a + b' a{a+b) l6 FUNCTIONS RATES AND DERIVATIVES. [Ex. II. dv 5. Given ay ■\- bx ■\- 2cx ■\- ab = o,\.q find -j-. , , ■' ^ ' ' ^x dy _ b + 2c dx ~ a * dy 6. Given y log a -^ x sin a —y cos a — « jir + tan a = o, to find ~. dy' a — sin a 7. Given ay cos'* a ~2b{i — s\vi(x)x — b{a — x cos^ a), to find ^ dx log rt — cos oc ' s^ a), to find ^. will vary with the value of Xy and the point will describe a curve. The tangent line to a curve is defined as follows : — The tangent to a curve at any point is the straight line which passes through the pointy and has the direction of the curve at that point!^ Hence, for any point of the curve, denotes the inclina- tion to the axis of x of the tangent line at that point. * It will be shown hereafter (Art. 49) that, in the case of the circle, this general definition of a tangent line agrees with that usually given in Plane Geometry. 20 FUNCTIONS RA TES AND DERI V A TIVES. [Art. 26. 26. Now, if a point, at first moving in the curve, should, after passing the point whose abscissa is a, so move that the rates -7- and —r retain the values which they had at the in- at at stant of passing the given point, the direction of its motion will become constant, and the point will describe a straight line tangent to the curve at the given point. The value of dx may be repre- sented by an arbitrary increment of X as in Fig. 3 ; the value of dy will then be represented by the corre- sponding increment which would be received by y, were the point moving in the tangent line, as indicated in Fig. 3. the diagram. Hence dy -y- = tan 0, ax which is evidently independent of the assumed value oidx.^ It follows that the value of the derivative of f(x), for any value of X, is represented by the trigonometric tangent of the inclination to the axis of x of the curve y =/(;ir), at the point corresponding to the given value of x, 27- The moving point, which is conceived to describe the curve, may pass over it in either of two directions differ- ing by 180°. The two corresponding values of give, how- ever, the same value of tan <^, since tan (^ ± 180°) = tan ^. Thus, in Fig. 3, the point P may be regarded as moving so as to increase x and 7, in which case both dx and dy will be positive, and ^ will be in the first quadrant ; or P may * In other words, the value of the derivative is determined by the form of the function/" which determines the curve, and the value ol x which fixes the position of/*. § III.] GEOMETRICAL MEANING OF THE DERIVATIVE. 21 move in the opposite direction, making dx and d^ negative, and placing in the third quadrant. In either case, ~ or tan(/) is positive. 28. It is evident that when f{x) is an increasing func- tion, as in Fig. 3, ~j~ is positive, and that when it is a de- . . . dy . creasing function, -r- is negative. Thus the sign of f {x) for any value of x is positive or negative according as f(x) is, for that value ot x, an increas- ing or a decreasing function. For example, it is evident that the value of the derivative of sin;ir must be positive when x is between o and ^tt, negative when x is between ^tt and |- t, and so on. When the notation —r- is used, the value of the derivative dx corresponding to a particular value ^ of ;ir is expressed by dy~\ ~ which is equivalent to/' (^). See Art. 2. Examples III. 1. If a point move in the straight line 2y — 7.r — 5 = o, so that fts ordinate decreases at the rate of 3 units per second, at what rate is the point moving in the direction of the axis of ^? dx^_6_ dt 7' 2. If a point starting from (o, b) move so that the rates of its co- ordinates are k and k', show that its path \s y= jnx -\- b, 7n being k' equal to 7- Express x and y in terms of t {Art. 14), and eliminate t. 3. If a point moving in a curve passes through the point (5, 3) 22 FUNCTIONS RATES AND DERIVATIVES. [Ex. III. moving at equal rates upward and toward the left, find the value of — ^ , also the equation of the tangent line to the curve at the given point. -^- = — I, and^ + ;f = 8. 4. If a point is moving in the straight line ;r cos a + J sin a = /, its rate in the positive direction of the axis of x being / sin ex., what is its rate of motion in the direction of the axis of ^? — / cos oc, 5. Given ay sin« — a;ir+ajr cosa — ^^ sec a = o; show that is con- stant and equal to \a. 6. U/{x) = tan;ir, show that/'(;r) must always be positive. 7. Show, by trac'ng the curve, that if j = x^, _Z can never be negative. CHAPTER 11. The Differentiation of Algebraic Functions. IV. The Square, 29. In establishing- the formulas for the differentiation of the simple algebraic functions of an independent variable, we find it convenient to begin with the square. The object of this article is, therefore, to express dix^) in terms of x and dx. We first deduce a relation between two values of the de- rivative of the function and the corresponding values of the independent variable ; for this purpose, we assume two values of the variable having a constant ratio m. Thus, if z^:^mxy ^' = m^ x^. Differentiating by equation (2), Art. 21, dz = m dxy and diz^) = n^ d{x^) ; dividing, we obtain d^z') d{x') — — = m — — . az ax Whence, dividing hy z — m x\o eliminate m^ we have _i d{^) _\_ d{x') s' dz ^ X ' dx (I) 24 ALGEBRAIC FUNCTIONS. [Art. 29. The derivatives -^- and -^ are, by Art. 23, functions of z and of x respectively, independent of the values of dz and dx\ moreover, equation (i) is true for all values of x and z, these quantities being entirely independent of each other, since the arbitrary ratio m has been eliminated. There- fore, either of these quantities may be assumed to have a fixed value, while the other is variable ; hence it follows that the value of each member of this equation must be a fixed quantity, independent of the value of x or of z. Denot- ing this fixed value by <:, we therefore write i ^(fl) _ x' dx ~^' or dix") = cxdx (2) 30. To determine the unknown constant c, we apply this result to the identity {x-\-/if = x' + 2hx + h\ Differentiating each member (Art. 22) by equation (2), we have c {x + h) d{x + h) =^ cxdx-h 2h dx ; since d{x + h) = dx^ this equation reduces to chdx =^ 2k dxy or {c — 2) h dx = o. Now, since k and dx are arbitrary quantities, this equation gives c = 2; this value of c substituted in (2) gives d{x'') = 2x dx (a) That is, t/ie differential of the square of a variable equals twice the product of the variable and its differential. § IV.] THE SQUAEE. 2$ 31. Employing the derivative notation, this result may also be expressed thus : — If /{^)=A f\x) = 2X, This derivative is negative for negative values of x, there- fore, for these values, x"^ is a decreasing function, as already mentioned (Art. lo) in connection w^ith the curve illustrating this function. Since X and dx are arbitrary, we may substitute for them any variable and its differential. Equation {a) therefore en- ables us to differentiate the square of any variable whose differential is known. Thus, — d{^x - if = 2(5^' - 3) sdx = io(5^ - 3) dx. Again, d{a x^ + d xf = liax" ^bx) d(a x' -\-bx) = 2{ax'' -f bx) {2ax + b) dx. The Square Root 32> To derive the differential of the square root, we put y = |/^, whence y^ ^= x] differentiating by {a), 2y dy = dx, . dx dy= ^ 2y y= ^x,.'. 41/-) = ^ (*) That is, tke differential of the square root of a variable is equal to the quotient arising from dividing the differential of the variable by twice the given square root. 26 . ALGEBRAIC FUNCTIONS. [Art. 32. Thus. 4^(.»_.=)3^_^-_^^, or, using derivatives, dx ^{a'-*x') Examples IV. '^ I. Differentiate {7.x + 3)*, and find the numerical value of its rate, when X has the vahie 8, and is decreasing at the rate of 2 units per second. The differential required is denoted by d\{7.x -\- 3)^], and the rate by 7; : the rzven rate -j— = — 2. dt ^ dt 152 units per second. 2. Find the numerical value of the rate of {x"^ — 2;r)% when x = y, and is increasing at the rate of ^ of one unit per second. Differentiate the given expression before siibstitutijig. ii units per second. V 3. Find the numerical value of the rate of 4/(7'' + x"^), whenj/ = 7 and ;r = — 7, if J is increasing at the rate of 12 units per second, and X at the rate of 4 units per second. 4 |/2 units per second. • 4. If f{x) = x — j^{x''- «'), find f'{x), and show that f{pc) is a decreasing function. _ £ / (^) - I - ^(^2_^2y •J 5. Differentiate the identity {\/x + \/ay = x + a ■\- 2 ^dx, and show that the result is an identity. 6. Differentiate y (f a _ 3^"^ )' 77/*? co7tstant factor ,, ^ -r. should be separated from the variable •' ^{a^ ~ 2ab) ^ -^ factor before differentiation. i x — a j^ia^ — 2a b) \/{x^ — 2ax) § IV.] EXAMPLES. 27 ^' 7. If fix) = (I + -r^)^ . f\x) = --^^-r. (I + xy X y/lO. li fix) = ^^ -, /'(^) = I + ,^ ., ,.^ Rationalize the denominator before differentiating. J x^ y* dy . ^ \i. Given — i + -7^ = i. express -j— in terms of x, and give the values r ^~l ^ dy-\ dy b X \J\i. Given y = 4^.r, express -7— in terms of x, also in terms oiy, and eive the values of -~ ^^a ^L . ^_.4/— = — . ' ^ 13. A man is walking on a straight path at the rate of 5 ft. per second; how fast is he approaching a point 120 ft. from the path in a perpendicular, when he is 50 ft. from the foot of the perpendicular.? Solution : — Let ;r denote the variable distance of the man from the foot of the perpendicular, so that -rr may denote the known velocity of the man, and let a denote the length of the perpendicular (120 ft.); then the distance of the man from the point is y'(a' + jr'^), of which the rate of change is denoted by d\ i^ia'^ -V xy^ X dx dt ~ ^{d' +^^) dt ' At the instant considered, x = $0 ft., while a = 120 ft., and —j- = — 5 f t 28 ALGEBRAIC FUNCTIONS. [Ex. IV. per second. By substituting these values, we obtain — i||. Hence his distance from the point is diminishing (that is, he is approaching it) at the rate of i|f ft. per second. ^ 14. If the side of an equilateral triangle increase uniformly at the rate of 3 ft. per second, at what rate per second is the area increasing, when the side is 10 ft. .^ * * 15 -^3 sq.ft. 15. A stone dropped into still water produces a series of continu- ally enlarging concentric circles ; it is required to find the rate per second at which the area of one of them is enlarging, when its diame- ter is 12 inches, supposing the wave to be then receding from the centre at the rate of 3 inches per second. ^ (§ y j^ jLyrt^ " y 16. If a circular disk of metal expand by heat so that the area A of each of its faces increases at the rate of o.oi sq. ft. per second, at what rate per second is its diameter increasing.? 1 . 1 ,:r-sv.'''^*nr^; ^ 17. A man standing on the edge of a wharf is hauling in a rope attached to a boat at the rate of 4 ft. per second. The man's hands "being 9 ft. above the point of attachment of the rope, how fast is the boat approaching the wharf when she is at a distance of 12 ft. from it.? 5 ft. per second. " 18. A ladder 25 ft. long reclines against a wall ; a man begins to pull the lower extremity, which is 7 ft. distant from the bottom of the wall, along the ground at the rate of 2 ft. per second ; at what rate per second does the other extremity begin to descend along the face of the wall.? 7 inches. 19. One end of a ball of thread is fastened to the top of a pole 35 ft. high ; a man holding the ball 5 ft. above the ground moves uniformly from the bottom at the rate of five miles an hour, allowing the thread to unwind as he advances. What is the man's distance from the pole when the thread is unwinding at the rate of one mile per hour } 1^6 ft. 20. A vessel sailing due south at the uniform rate of 8 miles per hour is 20 miles north of a vessel sailing due east at the rate of 10 miles an § IV.] EXAMPLES. 29 hour. At what rate are they separating — (or) at the end of i^ hours.? (3) at the end of 2^ hours ? Express the distances in terms of the time. {a) J-^V miles per hour. 21. When are the two ships mentioned in the preceding example neither receding from nor approaching each other ? Put the expression for their rate of separatioti equal to zero. When / = |o of an hour. 22. Derive, by the method employed in Art. 29 to determine the differential of the square, the result d\ — \ = — ~,c being an unknown constant. V. The Product, 33- Let X and y denote any two variables ; in order to derive the differential of their product, we express xy by means of squares, since we have already obtained a formula for the differentiation of the square. From the identity (x + j)' = x"" + 2xy +/' , we derive xy^^ix^-yy-^x'-^^f. Differentiating, d{x)') = {x + y) (dx + dy) — x dx — ydy, therefore, d{x y) ^= y dx -V x dy {c) Since x and j/ denote any variables whatever, and dx and dy their differentials, we can substitute for x and y any variable expressions, and for dx and dy the corresponding differentials. Thus, ^[(l +x') 4/(«' - ^')] = 4/(^> - x')2xdx - ^~~^f^ 2d — ;^x^ — I X dx. i/(d - x') 30 ALGEBRAIC FUNCTIONS. [Art. 34. 34. Formula {c) is readily extended to products consist- ing of any number of factors. Thus X^tx^ x^x^, . . . xp denote the product of / variable factors, then d(x^x^ x^ x^ — x^x^'"Xp dx^ + x^ d{x^ x\ • • -xj) < = x^x^' • -Xpdx^+XiX^' • 'Xpdx^ + XyX^d{x^' • -Xp) = x^x^' ' 'XpdXi+x^x^' ' 'Xpdx^' • • +XiX^' • 'Xp_^dxp. . {d) The Reciprocal, 35. The differential of the reciprocal may now be obtained by means of the implicit form of this function. Denoting the function by j/, we have Differentiating the latter equation by formula {c), we obtain ydx + xi whence dy=^ — ydx + xdy = o, ydx ~x"' substituting the value of /, 4)—^^ • •('^3 X Formula {d) enables us to differentiate any fraction of which the denominator alone is variable ; thus, Ja + b\ dx \a-\-x/ ^ Ha'\-xY § v.] THE QUOTIENT. 3^ The Quotient, 36> By the term quotient, as used in this article, we mean a fraction whose numerator and denominator are both variable. In deriving its differential, the quotient is re- garded as the product of its numerator by the reciprocal of its denominator. Thus, applying formulas (c) and {d\ M^ ii) dx X dy ■-y-'J-' ydx — xdy ■ yi y W It will be noticed that the negative sign belongs to the term which contains the differential of the denominator. As an illustration of the application of this formula, we have I2X — a\ __ 2{x'-^B) — 2x(2x — a) __ b-vax — x^ Formula (e) is to De used only zvhen both terms of the fraction are variable ; for, when the numerator is constant, the fraction is equivalent to the product of a constant and the reciprocal of a variable, and, when the denominator is constant, it is equivalent to the product of a constant by a variable factor. Thus, if it be required to differentiate the fraction , the use of formula (e) may be avoided by first making the transformation, x^^-a^ X a =- + -; ax a X ALGEBRAIC FUNCTIONS. [Art. 36. since, in this form, one term of each fraction is constant Hence, dx adx \ ax J ~~ a x^ ' The Power. 37- To obtain the differential of the power when the exponent is a positive integer, suppose each of the variables x^x^x^"'Xp in formula {c'\ Art. 34, to be replaced by x. The first member contains/' factors, and the' second/ terms ; the equation therefore reduces to d{x^)=px^'' dx (i) Next, when the exponent is a fraction, let y = xif then f = x^ ; differentiating by (i), / and ^ being positive integers, we have ^y ^ dy=px^~^dx, p x^~^ therefore, dy=-- ——j dx, q f Substituting the value of y. -. dx — -x'' ^ x^-\ ^ Again, when the exponent is negative, we have d(xi)= — ' dx=:-x'' dx (2) 1 x^-\ 1 ' -^ § v.] THE POWER. 33 Differentiating by formula {d), Art. 35, we obtain d{x )=— -^, and, since 7n is positive, we have, by (i) or (2), _ inx"^' ^ dx _ , r d{x '") = — -^, ——mx "• V/.r. . . . (3) Equations (i), (2), and (3) show that, for all values of w, d{xr) — nx''-^dx (/) By giving to 71 the values 2, J, and — i, successively, it is readily seen that this more general formula includes formulas {a), (b) and {d). 38. It is frequently advantageous to transform a given expression by the use of fractional or negative exponents, and employ formula (/) instead of formulas {b) and {ci). Thus, ^[(-^-2Py^] = ^(^' - ^^T' = S{a-- 2x')-'xdx, and d —-, -3 \=^d{a-\-x^-^ = — %(a + x) -^dx. When the derivative of a function is required, it may be written at oiice instead of first writing t-he differential, since the former differs from the latter only in the omission of the factor dx^ which must necessarily occur in every term. Thus, given -- ^^— r^ = ;ir(l-h;r')-^ we derive ^ = (i +^')-^ - \x{\ -^x")-'^ . 2x = ^f-^g 34 ALGEBRAIC FUNCTIONS. [Ex. V. J Examples V. 1. From the identity xy = \{x ■\- yY — \{x — yY derive the formula foi differentiating the product. ^ . « + bx + cx"^ ifferentiate . > 2. D X Put the expression in the form — + d + e x. ie rAdx. \ 3. Find the derivative of J = ^^rzrp' See remark. Art. zi, _ == (^» ^ ^.) ___^. xj 4. J = Sf{x^ - «'). 5-J^ = dy 3.r' dx ~ (a^ — .rO' * n/ 6. J/ = (I + 2jr^)(i + 4jr»). -£ = ^{i + 3x+ io;r»). ^^r ^ 8. ^ = (1 + .if (I + xy. ^= 4(1 + ^y (I + -^')(i + ■*• + 2^'). 9. J = (I + ;r"')" + (I + x")"*. ~- = mn[{l + ;r'»)*-'^-'«-^ + (l + jr^")-" -';»:«-»]. / ^5 _ 2a^ V 10. y = . / a —X dy = I + a^ dx {x-ar ±. a + X § v.] EXAMPLES. 35 12. y dx x' ^{x"" — ay y ab r^ y, o dy ab ix^ — a^ V U. ^ = ^-^ + ^— ^•. If = i[(i - -)-^ - (I 4- x)-ij. ^ l6. J = (« + -r)^ (^ — ;ir)*.r'. dv -J-=x{a + xy {b — xY \ia b -{■ {^b — 6d) x — gx""]. I ;r" 4- I dy 2nx'^-'^ i8. jj/ = (3^ + 2a,r)3 (^ ~ ax), '/' — — S^""-*" '^(3'^ + ^ax). v/19. {2a X — x'^y Put in the form {a" — b'''){7.ax — ;r') -*. ~£, = sC^' — P^ yj 20. y = V(^-' - -r') bx ^^' ^~ ^{2ax — xy /22.J=|/^. / — t/ '{2ax - ^')« • ^J. a' rt'^ {a^ _^')r dy n ^.r dx- (2a X I -:r')a- dx~ ii^x)^(i —xy 23. y = ^{a^ + x^) — X r> . ,^ , , dy I r a"" + 2X'' "1 fiattonahze the deno?mnator. -y- = --r- — ,, „ - — ^ -r 2x \. dx a^\_ \/{a^ + ^') J 36 ALGEBRAIC FUNCTIONS. [Ex. V. v 24. Two locomotives are moving along two straight lines of railway which intersect at an angle of 60° ; one is approaching the intersection at the rate of 25 miles an hour, and the other is receding from it at the rate of 30 miles an hour ; find the rate per hour at which they are separating from each other when each is 10 miles fro;ii the intersection. 2\ miles. \ ; 25. A street-crossing is 10 ft. from a street-lamp situated directly above the curbstone, which is 60 ft. from the vertical walls of the opposite buildings. If a man is walking across to the opposite side of the street at the rate of 4 miles an hour, at what rate per hour does his shadow move upon the walls — {pi) when he is 5 ft. from the curb- stone ? (/i) when he is 20 ft. from the curbstone ? / (a) 96 miles ; {p) 6 miles. 26. Assuming the volume of a tree to be proportional to the cube of its diameter, and that the latter increases uniformly ; find the ratio of the rate of its volume when the diameter is 6 inches to the rate when the diameter is 3 ft. ■^. 27. If an ingot of silver in the form of a parallelopiped expand lo^oo P^-rt of each of its linear dimensions for each degree of tempera- ture, at what rate per degree of temperature is its volume increasing when the sides are respectively 2, 3, and 6 inches ? If X denote a side, dx may be assumed to denote the rate per degree of temperature. -^-^ of a cubic inch. 28. Prove generally that, if the coefficient of expansion of each linear dimension of a solid is k, its coefficient of expansion in volume is T,k. Solution : — Let X denote any side ; then, if V denote the volume, we shall have V= cx^', c being a constant dependent on the shape of the body. Therefore dV = y x" dx ; or, since dx — kx, dV= zkcx^ = 3/^ V, CHAPTER III. The Differentiation of Transcendental Functions. VI. The Logarithmic Function. 39. In this chapter, the formulas for the difFerentiation of the simple transcendental functions are to be established. We begin by deducing the differential of the logarithmic function, employing the method exemplified in Art. 29. The symbol log;ir is used in this article to denote the loga- rithm of X to any base, and log^;ir is used when we wish to designate a particular base b. Let z = in X, differentiating by Art. 21 dz = vt dx, whence Multiplying hy z = mx, to eliminate m, we obtain h d{\ogz) _ d{\ogx) F^ dz -"" dx ^^^ r,>t 1 . . d(\o£[,z) ^diXoQ-x) , . . The derivatives, , and , — , are, by Art. 23, func- log z =z log m + log;ir, t. 21, and d{\ogz) dz - d{\ogz) = d(}ogx)\ d(\ogx) m dx 38 TRANSCENDENTAL FUNCTIONS, [Art. 39. tions of z and of x respectively, independent of the values of dz and dx\ moreover, equation (i) is true for all values of X and ^, these quantities being entirel}^ independent of each other, since the arbitrary ratio m has been eliminated. Hence, in equation (i), one of the quantities, x or z, may be assumed to have a fixed value, while the other is variable ; whence it follows that the members of this equation have a fixed value independent of the values of x and z ; we therefore write ^(l02:;ir) , . X T^ — - = a constant (2) dx ^ This constant, although independent oi x, may be dependent on the value of the base of the system of logarithms under consideration. Denoting the base of the system by b^ we therefore denote the constant by B^ and write equation (2) thus, — 41og,^') = — T- (3) 4-0. To determine the value of B, we establish a relation between two values of the base and the corresponding values of this unknown quantity. Denoting another value of the base by a, and the corre- sponding value of the unknown constant by A^ we have A^Oga^) = -^ (4) The relation sought may now be obtained by differentiat- ing, by means of (3) and (4), the identical equation log^;f = log^^ log^;ir,* (5) * This identity is most readily obtained thus, — by definition § VI.] THE LOGARITHM. 39 Adx , ,Bdx thus obtaining — — = log^ b — — , X X or BXoZab^A, hence ^^Za^^ ~ ^» that is, A is the logarithm to the base a of h^ ; whence we have b^ ^ a^ (6) Now, it is obvious that the value of a^ cannot depend upon b, hence equation (6) shows that the value of b^ likewise cannot depend upon b\ b^ must, therefore, have a value entirely independent of b. Denoting this constant value by e, wc write 3^ == £ (7) Adopting this constant as a base, and taking the loga- rithms of each member of equation (7), we have Blo^^b = I, I whence B \og,b' Introducing this value of B in equation (3), we obtain In this equation, the differential of a logarithm to any given base is expressed by the aid of the unknown constant e, 41. The constant e is employed as the base of a system of taking the logirilhra to the base a of each member, we have logax = logbx loga I tan«/) < i, and when ^ < i tan > i ; the curve, therefore, lies below this tangent, as shown in Fig. 4. The point (e, i) is a point of the curve ; let j5. Fig. 4, be this point, then OR will represent the Napierian base, and B R = I. Since OA =1, and AR> BR, OR>2', that is, the Napierian base e is somewhat greater than 2. The quantity e is incommensurable : the method of com- puting its value to any required degree of accuracy is given in a subsequent chapter. Logarithmic Differentiation. 43, The differential of the Napierian logarithm of the variable ,r, that is the expression -^ , is called the loga- rithmic differential of x. When X has a negative value, the expression log;r has no real value; in this case, however, log(— ;r)is real, and we have / XT d{—x) dx ^[log (- ^)] = ^ = — -. 42 TRANSCENDENTAL FUNCTIONS. [Art. 43. This expression therefore, in the case of a negative quantity, is identical with the logarithmic differential of the positive quantity having the same numerical value. 44. The process of taking logarithms and differentiating the result is called logarithmic differentiation. By means of this method, all the formulas for the differentiation of alge- braic functions may be derived^ * In the following logarithmic equations, it is to be under- stood that that sign is taken in each case which will render the logarithm real. By differentiating the formulas, — log {±xy) = log {±x) + log (± j), log(±,r«) = ;/log(±;t'), d(xv) dx dy we obtam = — + -^, • xy X y^- X \yJ X y d{x") dx — '^— ^^ — ■• X"' X These formulas are evidently equivalent to (r), (r), and (/), of which we thus have an independent proof. 45. The method of logarithmic differentiation may fre- quently be used with advantage in finding the derivatives of complicated algebraic expressions. For example, let us take 71 = r (I) Hence, we derive § VI.] THE EXPONENTIAL FUNCTION. 43 log- « = -J log {2X) + i log (l — ^') - I log {x — 2), . . (2) differentiating, du _ _j 3 x_ 2 _ J__ 11 dx ~ 2x ^ I — y ^ X — 2 (3) adding and reducing, du — 8;r' + 24;tr' — .r therefore udx 6 (i — y') (^.r — 2)jir du — 8,r' + 24jr" — x — 6 dx ^{2xf{i-x'f{x-2f' For certain values of Xy one or more of the quantities whose logarithms appear in equation (2) become negative. When this is the case these logarithms should, strictly speaking, be replaced by the logarithms of the numerical values of the quantities in question ; this change however would not affect the form of equation (3). See Art. 43. Exponential Functions, 4-6. An exponential function is an expression in which an exponent is a function of the independent variable. The quantity affected by the exponent may be constant or vari- able. In the first case, let the function be denoted by J = ^-^ (i) \i a is negative, a' cannot denote a continuously varying quantity. We therefore exclude the case in which a has a negative value, and regard a^ as a continuously varying pos- itive quantity. Taking Napierian logarithms of both members of equation (i), we have log J = X log^ ; differentiating by (^0. 44 TRANSCENDENTAL FUNCTIONS. [Art. 46. — = loGT a . dx ; y ^ ^ hence dy ^=z loga.ydjVj or d^a"^) = loga.a'^dx {/t) Exponential functions of the form e-*" are of frequent occur- rence. Putting ^ = € in formula (//), we have * d{e^) = e^dx; (//) hence the derivative of the function e^ is identical with the function itself. This function is the inverse of the Napierian logarithm ; it has been proposed to denote it by the symbol exp X. 4-7. When both the exponent and the quantity affected by it are variable, the method of logarithmic differentiation may be employed. Thus, if the given function be we shall have log-s" = x"^ log {nx) ; differentiating, — = x"" — + 2x log {n x) dx, hence d\{n xy'\ = {n x^ x[i + 2 log (n x)] dx. 1^^, and Examples VIl -^ I. Given the function j/ — logj;r; show that — dx hence prove that the tangent to the corresponding curve, at the point whose abscissa is e, passes through the origin. Put a =. X ^=it in equation 5, Art. 40. ] VI.] EXAMPLES. ■ 45 ^ 2. y = ;r" log X. -^ = jr" ~ ^ (I + « log x), ax / . ., . dy \ 3. j = log (log ^). -^ — dx X log jr 4. jF = log[log(^ +<^^)]. - — dx {a + <^^") log(^ + bx"") 'i 5. >/ = 4/.r-log(4/^+ I). -^ dx 2 ( y'jf 4- i) •^ ^ 4/.^ — |/^ ^/.r (^? — x) j^x Put in the form^ log ( 4/^ + ^/^r) — log ( y'<^ — |/.r). ^ 7.^ = logfV(.-.)4-V(--.)]. _^^ = _-__i___. 4 8.^ = log [.r + 4/(.r^ ± ^^)]. -^ V 9. J = log—- — - dx ^{x-' ± a^) dy _ I -/(I + x"") dx x{i + x"") ^io.y= iogyO+^)_+4/(i-^) ^ _ 4/(1 4- ■^') — |/v I — -^) ^-^^ X 4/(1 — jr'^) ^11. y = lot; r^ + i/Oi- — x')\ -4- = ., .. ----„-. ^ , „ y 12. ^= log //or 4/(^'^ — x'') \_x + 4/(rt^ — A'^)]' ^/ l/C^'-* + d')—x dx X ^{x'' + «■■')' dy i 1 , , « (2Ji: — ^) ^ ^r*^ + = — sin 0, and — = dO, Substituting these values in equations (2), we obtain d{sind) = cosddd, (/) and d{cosO) = — sinddd (y) TAe Tangent and the Cotangent, 51. The differential of tan0 is found by applying formula (e) to the equation sin0 tan Q = ; cos Q ' . , r /. ^N cos <^(sin 0) — sin <^(cos &) thus, ^(tan0) = ^ ^ — ^-^ ^^ ^, ^ ^ cos w or ^(tan0) = ^^^z=sec'0^a {k) *In Fig. 5, dx is negative ; but, being in the second quadrant, cos0 is lil^ewise negative. 50 TRANSCENDENTAL FUNCTIONS. [Art. 5 1. The differential of cot is found by applying formula {k) to the equation COt0 = tan(i;r — Q)\ whence d{c,oX.Q) — — -^^^ = — costd'ddd, . . . (/) The Secant and the Cosecant. 52. The differential of sec is found by applying formula (^) to the equation sec^ = cos^ , ,, , sine do whence d^sec d) = -^^-^ = sec d tan Odd. , . (m) The differential of coseca is found by applying formula (m) to the equation cosec d = sec (i ^ — 6) ; , 7/ ^N cos Odd whence d {cosec d) = ^-^— - = — cosec cot 0^(?. . (n) sin u ^ ■^ The Versed-Sine. S3. The versed-sine is defined by the equation vers (9=1— cos Q \ therefore ^(vers (9) = sin a ^0 f^) § VII.] EXAMPLES, 51 Examples VII. 1. The value of ^^(sin^) being given, derive that of d?(cosO) from the formula cosO = sin (^TT — 0) ; also from the identity cos^G = I — sin^Q. 2. From the identity sec'^= i + tan'0, derive the differential of secO. 3. From the identity sin2 = 2 sin cos 0, derive another by taking derivatives. cos 2 = cos' — sin' G. 4. From the identity sin (0 ± \t:) = i |/2 (sinG ± cosG), derive an- other by taking derivatives. cos (G ± i tt) = i |/2 (cos G ^ sin G). 5. Prove the formulas : — ^/(log sin G) = — ^(log cosec 0) = cot G d^ ; //(log cos G) = — ^(log sec G) = — tan G d^ ; / 1 5. J = tan^* + log (cos^ 9). ^ 16. _y = log (tan + sec 0). 17. J = logtan(i7r + i0). ^ 18. _>/ = ;(:+ log cos (iTT—x). ■; 19. 7 = log yCsin x) + log ^/(cos ;ir). 20. J = sin n (sin 0)*. J sin;r ^ 21. J = I + tan X \ ' 22. J = t^'^cosbx, 4 _ Mcosx — b^\nx 3- J — og |/ ^cos^ + bsinx' dy ^0 = sec* 0. = e* cos e*. ■J- = sin ;r'' + ^x"^ cos ;r'. -^ = log a . «""* COS JIT. | = 2tan30. ^0 = sec 0. dy _ I ^""cos0* <^_ 2 ~dx~ \ ■\- tan y ^;r = cot 2X. dv jQ=n (sin 0)" - ' sin (?z + I) 0. dy cos^x — sin'^r dx~ (sin X + cos;*:)''* dy -^ = ef'' {a cos b X — bsmbx). -ab dy^ dx d^ cos^ X — b^ sin' X* § VII.] EXAMPLES. 53 ^ 2^. y ~ ^' (q.o's.x —s\T\x). -^= — 2fc^sinx 25. The crank of a small steam-engine is i foot in length, and revolves uniformly at the rate of two turns per second, the connect- ing rod being 5 ft. in length ; find the velocity per second of the piston when the crank makes an angle of 45° with the line of motion of the piston-rod ; also when the angle is 135°, and when it is 90°. Solution : — Let a, b, and x denote respectively the crank, the connecting-rod, and the variable side of the triangle ; and let denote the angle be- tween a ?Lnd X. ^. . We easily deduce * ^*^tf £^ ;ir = «cos0 + i/(^' — rt'sin'»0); >tc^^' (^--^•^^ etc whence dx I ^'sinOcosQ \d^ dO In this case, -tj = ^n, a = i, and ^ = 5. When 6 = 45°, ;^7 = -f- ft. 26. An elliptical cam revolves at the rate of two turns per second about a horizontal axis passing through one of the foci, and gives a reciprocating motion to a bar moving in vertical guides in a line with the centre of rotation : denoting by 6 the angle between the vertical and the major axis, find the velocity per second with which the bar is moving when = 60°, the eccentricity of the ellipse being \, and the semi-major axis 9 inches. Also find the velocity when 6 = 90°. The relation between and the radius vector is expressed by the equation a(i — e^) r^'-^'^ 1 u ^cos9 9 When = 60°, -r- = — 12 4/3 TT inches. 27. Find an expression in terms of its azimuth for the rate at which the altitude of a star is increasing. Solution : — Let // denote the altitude and A the azimuth of the star,/ its polar distance, / the hour angle, and L the latitude of the observer ; the formulas of spherica! trigonometry give 54 TRANSCENDENTAL FUNCTIONS. [EX. VII. sin ^ = sin Z COS /^ + cos Z sin/ cos /, . . . . (i) and sin/ sin / = sin ^ cos h. ...... (2) Differentiating (i),/ and Z being constant, cos "--f.^ ~ COS Z sm/ sm /, (It whence, substituting the value of sin/ sin /, from equation (2), -— = — cos Z sin A. at It follows that — is greatest when sin^ is numerically greatest ; that is, when the star is on the prime vertical. In the case of a star that never reaches the prime vertical, the rate is greatest when A is greatest. VIII. The Inverse Circular Functions. S^. It is shown in Trigonometry that, if X = sin Qy the expressions 2n7t-\-0 and (2;^+ i) tt — <9, . . (i) in which n denotes zero or any integer, include all the arcs of which the sine is x\ hence each of these arcs is a value of the inverse function Among these values, there is always one, and only one^ which falls between — \n and +i7r; since, while the arc § VIII.] INVERSE CIRCULAR FUNCTIONS. 55 passes from the former of these values to the latter, the sine passes from — i to + 1 ; that is, it passes once through all its possible values. Let 0, in the expressions (i), denote this value, which we shall call the primary value of the function. 65- In a similar manner, if X = cos 9j each of the arcs included in the expression 2n7r ±e (2) is a value of the inverse function cos " ' ,v. One of these values, and only one, falls between oand n ; since, while the arc passes from the former of these values to the latter, its cosine passes from + i to — i ; that is, once through all its possible values. In expression (2), let d denote this value, which we shall call the primary value of this function. 56, In the case of the function cosec ~ ^ x^ the definition of the primary value that was adopted in the case of sin~*;ir, and the same general expressions (i) for the values of the function, are applicable. In the case of the function sec~*;tr, the definition of the primary value adopted in the case of 56 TRANSCENDENTAL FUNCTIONS. [Art. 56. cos-'-JT and expression (2) for the general value of the function are applicable. Finally, in the case of each of the functions tan~';ir and cot~';i; t\iQ primary value {d) is taken between —^7t and -\-^7t, and the general expression for the value of the function is njt + e (3) The Inverse Sine and the Inverse Cosine, 67- To find the differential of the inverse sine, let Q = sin-^;tr; then x=sm8, and dx = cosOdd^ dx or dQ= — ^. cos^ (I) If B denotes the primary value of this function ; that is, the value between — ^7rand+|-^, cos0 is positive. Hence the upper sign in this ambiguous result belongs to the differential of the primary value of the function ; it is therefore usual to write 4jr 4. J = sin - ' {2x^), — 5. ^ = sin-* (cos;r), 6. y = sin (cos" *;r), 77" 7. ^ = sin-* (tan;r). 8. / = cos-* (2Cos;r). 9. j=;rsin-*;r+ 4/(1 — ^r"). 10. _y = tan-*f*. 11. y={x^ + i) tan-*;r — ;r. ^ V 12. ^=«''sin-*|^ + ;r v'C^'^ - ^r'^). ^ = 2 |/(^^ - jr") ^ V 13. ^ = tan-*j^^;-^. ^+ I dx- . ^(1-4^*)- dy dx^" ^' dy dx~ X v{i-^r dy sec'jr dx 4/(1 [ — tan'^^r)* ^^ 2sin-r ^-r --^./(i. — 4COS='^)* ^J //;»: = sin-* jr. dy dx I -£-+£-"' dy dx- 2;rtan-*;ir. [4. y = sin • 1 — dy m{\ + -r^) dx~ - i^{m^- -2);r» + ^* dy I 4/2 ' dx ^{i—2x—x*y 62 TRANSCENDENTAL FUNCTIONS. [Ex. VIII. *> V — Lau 4/(1-^^)- i6. ^ = sec- V i/i7. y^ sin- -1 v/. 18. y = sin- ■» 4/(sin r). v/. 19. y = 4/(1 — x')sin->jr— jr. , »2 4- ^ 20. _y = tan-^- mx I -x» -^ , /I — cosjtr 22. y = tan - ^ 4/ — ; . ■^ r I + cos;r J J 1/ jjrsin-';r , , ^ 24. y—{x-\-d) tan-' |/ 4/(«;f). dx~ 4/(1 — ^^)' ^ I dx ^{cr-x'^y dy a dx~ a" ■Vx''' -^ = 1^(1 +cosec^). dr dx ;rsir l-i^ir ^(i -^^) ^ ^/;r I I ^__ 2 ^J Tx-^' ^/ sin -';r ^x- ~(.-x')«- ^7 dx- '^""'/l- IX. Differentials of Functions of Two Variables. 64. The formulas already deduced enable us to differen- tiate any function of two variables, expressed by elementary functional symbols ; the application of these formulas is, how- § IX.] FUNCTIONS OF TWO VARIABLES. 63 ever, sometimes facilitated by a general principle which will now be shown to be applicable to such functions. The formulas mentioned above involve differential factors of the first degree only. It follows, therefore, that«the differ- entials resulting from their application consist of terms each of which contains the first power of the differential of one of the variables. In other words, if du =^{x,y)dx-V\\){x,y)dy (l) Now, if y were constant, we should have ^ = o, and the value oi dii would reduce to that of the first term in the right- hand member of (i); hence this term may be found by differ- entiating u on the supposition that y is constant^ and in like manner the second term can be found by differentiating u on the supposition that x is constant. The sum of the results thus obtained is therefore the required value of du, 65. As an example, let u^. Were v constant, we should have for the value of dz, by formula (/), Art. 37, vu'"—'^du\ and, were u constant, we should have, by formula (Z^), Art. 46, log u . u"^ dv] whence, adding these results, ds ^ «" - \v du-\-u log ti dv\ 64 DIFFERENTIATION. [Art. 65. Although this result has been obtained on the supposition that u and v are independent variables, it is evident that any two functions of a single variable may be substituted for u and V. Tiius, if u = nx and z/ = 4r', we have z = {n x^y * and, on substituting, ds = {n xy -'^{x'^ndx+ n x log {n x) . 2x dx), = x(nxy[i-\-2\og{nx)']dx, which is identical with the expression obtained in Art. 47, for the differential of this function. Examples IX. I. u = xy e* + «y. du=e'+^yly(i + x)^;r + x(l + 2y)dy]. 2.u = log tan^. du=2^-^^^^Z^. i/^sin2 — y 3. « = logtan-':^. ^^_ ^^^-^- -^^J y C^'^+jOtan-^^ ^^^*^-±^^y ^^^- b--y--2 4/(.r7)1 ^ydx ^\ x-y-^2 ^/{xy)\ i/xdy ^V{.xy){x^yy J J r ^^ ""y ■^■^_ y'^^ . x{xdy-ydx)^ § IX.] MISCELLANEOUS EXAMPLES 65 y / dy _ dx X + 2 2(1 + -)' dy (^' -b'')x dx {a^-x\ )'^(^*^-. .')* Va{Vx- ■ Va) / / ' 9. Given x = r cos 9, and j = r sin Q; eliminate 6 and find dr ; also eliminate r and find dB, . X dx -{■ y dy , .. xdy—ydx dr = .. , , .;. , and dO = — 3 , -^ ^ . Miscellaneous Examples. ^' -^ V + Vx' dx 2 Vx V(a + x)(Va + Vx)" ' , 4.,= (Vx-2Va)V{Va+Vx). |= ^^(^,%^,) . ^ _ (o:- i)(g' 4- i)e' dy _ £' {x s""' - 2x s^ + 2s' — x) ^ 6. ^ = log-^^ fr + i tan ^x. -f- = 7 — ■ TT — ^ — Tx • ■^ ^ (i + ^)« dx {i+x){i-^x) / 2sin~*^ , , I— ^ 4^ 2Jtsin"^^ 66 DIFFERENTIATION. [Ex. IX, ^ V 1 1. JF = ^,log ^ -' — V [a — X dy_ V (g" - x^) dx~ ^ X \ (i — :v'^)8sm~* 12. V = -'^ '- X X dy 1— X 1 -{• 2x^ ., 2\ • -1 ~ = ' i — Vii — X) sm ^x, ax X X ^ ^ n) ., .-w./ ^-^Q^^ ^^ 13. J log/f^ cos:^? dx sinjc* 4 I4.7 = tan-r^^.tanf]. dx~ 2{a + b cos jc) ' J _t I /^ 2 ^ 15.7= sec ^ — ^ . -f = — — ^ 2^—1 d!'.;C 1/ (l — ^') J 16. J = COS ^ -^ ^ _ 2nx'' X'" +1 ^- ;v'" + I dy_ 3c_ 'dx ~ V [b' -{a- xy^ • Jo -1 i/i— -^ dy \/ (1 — x) ^ 18. y = cos^jt: — 2i/ . ^ — JL^ J. ^ ^ i+x dx (^j^x)^ £/j^ logarithmic differentials. CHAPTER IV. Successive Differentiation. X. Velocity and Acceleration, 66. If the variable quantity x represent the distance of a point, moving in a straight Hne, from a fixed origin taken on the line, the rate of x will represent the velocity of the point. Denoting this velocity by v^ we have, in accordance with the definition given in Art. 17, dx , . "'^w ('> In this expression the arbitrary interval of time dt is re- garded as constant, while dx, and consequently Vj^, is in gen- eral variable. Differentiating equation (i) we have, since dt is constant, dt The differential of dx, denoted above by d{dx)y is called the second differential o{ X ; it is usually .written in the abbreviated form d^'x, and read " d-second xJ' The rate of Vx is therefore expressed thus : — dvjc _d^x It (dlf' 68 SUCCESSIVE DIFFERENTIATION. [Art. C}6, The rate of the velocity of a point is called its acceleration^ and is usually denoted by a ; hence we write the marks of parenthesis being usually omitted 4n the denomi- nator of this expression. 67. When the space x described by a moving point is a given function of the time /, the derivative of this function is, by equation (i), an expression for the velocity in terms of /. The derivative of the latter expression, which is called the second derivative of x, is therefore, by equation (2), an expres- sion for the acceleration in terms of /. A positive value of the acceleration a mdicates an algebraic increase of the velocity v, whether the latter be positive or negative ; and, on the other hand, a negative value of a indi- cates an algebraic decrease of the velocity. 68. As an illustration, let x denote the space which a body falling freely describes in the time /. A well-known mechanical formula gives ^ = W' 0) dx Hence we derive Vx—-r=gti (2) ai J dvx d'^x , X ^"'^ '^'=-^=^=^- ^3) In this case, therefore, the acceleration is constant and posi- tive, and accordingly v^^ which is likewise positive, is numeri- cally increasing. 69. When the velocity is given in terms of x, the acceleration can readily be expressed in terms of the same variable, as in the following example. § X.] VELOCITY AND ACCELERATION. 69 Given Vx—2^mx\ dvx dx whence -7— = 2 cos x -j- ; at at that is, ^'^ = 2 cos x,Vx = ^ cos jr sin ;ir = 2 sin 2x, The general expression for a^, when v^ is given in terms of x^ is _ dvx _ ^-2^^ ^^ _ dvx _ I ^(t^x) / \ ^~~ dt dx dt ~ ^ dx ' 2 dx Component Velocities and Accelerations, 70. When the motion of a point is not rectilinear but is nevertheless confined to a plane, its position is referred to co- ordinate axes ; the coordinates, x and y, are evidently functions of /, and the derivatives —- and -— , which denote the rates dt dt of these variables, are called the cojnponent or resolved velocities in the directions of the axes. Denoting these component veloci- ties by Vx^nd Vyy we have dx . dy t.. = ^, and v,=f^. Again, denoting by s the actual space described, as measured from some fixed point of the path, s will likewise be a function ds of /, and the derivative -y will denote the actual velocity of at the point. (Compare Art. 48.) Now, the axes being rectangu- lar, and denoting the inclination of the direction of the mo- tion to the axis of x, we have dx = ds cos ^, and dy = ds sin 0. -, dx ds . A dy ds , J. 70 SUCCESSIVE DIFFERENTIATION, [Art. /Q or Vx — V COS ^, and Vy =v sin ^. Squaring and adding, The last equation enables us to determine from the component velocities the actual velocity in the curve. 71. If we represent the accelerations of the resolved mo- tions in the directions of the axes by ajc and ofy, we shall have, by Art. 66, a.= ^ and ., = ^ . These accelerations, a^ and a^^ will be positive when the re- solved motions are accelerated in the positive directions of the corresponding axes ; that is, when they increase a positive re- solved velocity, or numerically decrease a negative resolved velocity. Examples X. J I. The space in feet described in the time / by a point moving in a straight line is expressed by the formula ^ = 48/ — 16/"; find the acceleration, and the velocity at the end of 2 J seconds ; also iind the value of t for which z; = o. Of = — 32 ; z; = o, when / = \\. J 2. If the space described in / seconds be expressed by the formula jf = 10 log 4 + i' find the velocity and acceleration at the end of i second, and at the end of 1 6 seconds. When t= i, v= — 2 and « = f . § X.] EXAMPLES. 71 , 3. If a point moves in a fixed path so that s= Vt, show that the acceleration is negative and proportional to the cube of thp velocity. Find the value of the acceleration at the end of one second, and at the end of nine seconds. — :^, and — jj-^. "^ 4. If a point move in a straight line so that x = a cos ^Ttty show that a= — ^Tt^x. V 5. If x = a£' ■{■ d€-\ prove that a = x. J 6. If a point referred to rectangular coordinate axes move so that X = a cos / + <^ and y = a sin f ■}■ c, show that its velocity will be uniform. Find the equation of the path described. Eliminate t from the given equations. V 7. A projectile moves in the parabola whose equation is y = x\.zxia —f — — x^, 2 V cos Of (the axis of _y being vertical) with a uniform horizontal velocity Vx^= V cos a ; find the velocity in the curve, and the vertical acceleration. v= V{V' — 2gy), and a, = —g, 8. A point moves in the curve, whose equation is x^ + ^f = a^^ 7'2 SUCCESSIVE DIFFERENTIATION-. [Ex. X, SO that Vx is constant and equal to k ; find the acceleration in the di- rection of the axis oiy. ^1^2 y 9. If a point move so that v — V{2gx); determine the acceleration. C/se equation (i), Art. 6g. ^ = g- \J 10. If a point move so that we have v"^ = c — M log x, n. yj II. If a point move so that we have determine the acceleration. a = — — 2X 2^ determine the acceleration. a — -. {x' + b')^ 12. The velocity of a point is inversely proportional to the square of its distance from a fixed point of the straight line in which it moves, the velocity being 2 feet per second when the distance is six inches ; determine the acceleration at a given distance s from the fixed point. - ~, feet. 2S y .... 13. The velocity of a point moving in a straight Ime is m times its distance from a fixed point at the perpendicular distance a from the straight line ; determine the acceleration at the distance x from the foot of the perpendicular. ol = m^x. V 14. The relation between x and / being expressed by f 1/ -^= \/{ax — x) —^ayeis "^ — ; find the acceleration in terms of x. oc= 5 • X \ 15. A point moves in the hyperbola / =P'^X' + q' in such a manner that Vx has the constant value c ; prove that § X.] EXAMPLES. 73 and thence derive ay by equation (i), Art. 69. 7 decreases algebraically throughout, the curve appearing convex when viewed from above. Fig. 7. 74. A point which separates a concave from a convex portion of a curve is called a point of inflexion, or 2, point of contrary flexure. It is obvious from the preceding article that, at a point of inflexion, like P in Fig. 8, f"(x) must change ugn ; hence at such a point, the value of this derivative must become either zero or infinity. 75. When a curve is described by a moving point, the character of the curvature is depen- dent upon the component accelerations of the motion. For, if we put Vx = r, or dx •=€ dt^ c denoting a constant, we have Fig. 8. § XI.] THE SECOND DERIVA TIVE. ^% and hence /"W = ?-^=f- Whence it follows that, if Vx is constant, ay and f"{x) have the same sign, and consequently that a portion of a curve which is concave when viewed from above is one in which ciy is positive when ax is zero. Successive Differentials, 76. The successive differentials of a function of x involve the successive differentials of x ; thus, if we have dy = ^x^dx, dy = 6x(dxy-\- 3;rW, and dy = 6{dxy + iSx dxd'x + sx'd'x. In general, if dy=/Xx)dx, dy=/"{x){dxy+/Xx)d''x, and dy =/"'{x) {dxy+ s/"{x)dxd'x +/'{^)d'x. Equicrescent Variables. 11, A variable is said to be equicrescent when its rate is con- dx stant ; since dt in the expression — - is assumed to be constant, dt dx is also constant, when x is equicrescent. In expressing the differentials of a function, it is admissible 76 SUCCESSIVE DIFFERENTIATION. [Art. 77. to assume the independent variable to be equicrescent, since the differential of this variable is arbitrary. This hypothesis greatly simplifies the expressions for the second and higher dif- ferentials of functions of x^ inasmuch as it is evidently equiva- lent to making all differentials of x higher than the first vanish. Thus, in the general expressions for d'^y and d^y given in the preceding article, all the terms except the first disappear, and it is easy to see that, in general, we shall have when X is equicrescent. 78. From the above equation we derive dx"" -^ ^ ' The expression in the first member of this equation is the usual symbol for the n\h derivative of y regarded as a function of x. The n\h. differential which occurs in this symbol is always un- derstood to denote the value which this differential assumes when the variable indicated in the denominator is equicrescent. The symbol —- is frequently used to denote the operation dx of taking the derivative with reference to ;r, and similarly the / d\'' d" symbol ( ^7- ) , or — -- , is used to denote the operation of tak- ing the derivative with respect to x, n times in succession. Examples XL V I. Find the second derivative of sec x, and distinguish the concave from the convex portions of the curve y ~ sec x. Also show that the curve y = log x is everywhere convex. § XL] EXAMPLES. 77 y 2. Find the points of inflexion in the curve j/ = sin x, y 3. Find the point of inflexion of the curve y =z 2X^ — ^^x"^ — I 2Jt: + 6. The point is (J, — J). ^ 4. Show that the curve y = tan x is concave when y is positive, and convex when y is negative. V 5. Find the points of inflexion of the curve y = X* — 2X^ — l2Jt:^ + 11^ + 24. The points are (2, — 2) and (— i, 4). / 6. If/W=i±|,findrW. /V) = (7^^e. yl 7. If/(^) =-, find/ (x). f {x)=-- ^^, . ^ 8. If j^ is a function of x of the form Ax"" -h Bx""^ + • • • + Mx + Ny prove that -j^ = i. 2. 3 • • • ;2 ^. UrX ^ 9. If/(^) = nfind/M^). v/ 10. If/ [x) = x' log (w^), find/^^ (^). / II. If/ (x) = log sin jc, find/'" (:r). 1^ 12. If/ {x) = sec ^, find/" (x) and/'" (.:«:). /" (^) = 2 sec^ ^ — sec x, and/"' (^) = sec x tan ^ (6 sec^:*: — i). / 13. If/ (x) = tan X, find/"' {x) and/'" (^). /'" (jc) = 6 sec^jc — 4 sec-^, and/'" (;t) = 8 tan .a; sec^^i; (3 sec^^^c — i). •'(*) = a'(losdri>". ri^)=i. /" , 2 cos X ^ ' sin' ^ * 78 J 18. SUCCESSIVE DIFFERENTIATION, [Ex. XI. If/ W = x% find/" (jc). /" (^) = ^' (i + log xf + :^-\ If^ = £^,findg. If^=e-=^^,findg. Ifj; = log(£^ + f-*),findg. If J dx'~ °(£-+f-y I ^ .d'y .d'y V 19. V 20, V 23. If _y = sin ' jv, find -y^. d''y _ gx + 6.v' If^ =£"»', find g. ^3 -T=i = — £^^°* COS ^ sin X (sin jc + 3). dx li y = , find -^ d'^y __ I — logjc ^JC"* ^ (i + log^)^ I + log X * ^/jJC^ Find the value of ^^(g""), when x is not equicrescent. d\e') =z e'{dxy + 2,^' d'x dx + £' d'x. 73 Find the value of -z-^ (sin 6), Q being a function of /. df' , . . /doY . do d'o d^O (sine) = - cose (^-j _ 3 sme - • ^ + cose — .. CHAPTER V. The Evaluation of Indeterminate Forms. XII. Indeterminate or Illusory Forms. 79. When a function is expressed in the form of a fraction each of whose terms is variable, it may happen that, for a cer- tain value of the independent variable, both terms reduce to zero. The function then takes the form - , and is said to be .o indeterminate, since its value cannot be ascertained by the ordi- nary process of dividing the value of the numerator by that of the denominator. The function has, nevertheless, a value as determinate for this as for any other value of the independent variable. It is the object of this chapter to show that such defi- nite values exist, and to explain the methods by which they are determined. The term illusory form-^s often used as synonymous with indeterminate form, and these terms are applied indifferently, not only to the form - , but also to the forms — , co- o, co — oo, O 00 and to certain others whose logarithms assume the form oo-o. When a function of x takes an illusory form for x—a, the cor- responding value of the function is sometimes called its limits ing value as x approaches the value a. 80. The values of functions which assume illusory forms may So EVALUATION OF INDETERMINATE FORMS. [Art. 8o. sometimes be ascertained by making use of certain algebraic transformations. Thus, for example, the function a — V{a^ - bx ) X takes the form - when x = o. o Multiplying both terms by the complementary surd a + V{a^ — bx\ bx b we obtain x\a + ^/{d' - bx)] a + V(«' - bx) ' The last form is not illusory for the given value of x, since the factor which becomes zero has been removed from both terms of the fraction. The value of the fraction for x = o is evi- dently — . 2a The following notation is used to indicate this and similar results ; viz., a - V{a' - bx)-] _ b ']: 2a the subscript denoting that value of the independent variable for which the function is evaluated. Evaluation by Differentiation, 81. Let - represent a function in which both u and v are u functions of ;tr, which vanish when x =■ a\ in other words, for this value of x, we have u = Of and v = o. § X 1 1 .] EVAL UA TION B Y DIFFERENTIA TION, 8i Let P be a moving point of which the abscissa and ordinate are simultaneous values of u and v {x not being represented in the figure) ; then, de- noting the angle POU hy 6, and the inclina- tion of the motion of P to the axis of u by ^, we have dv Fig. 9. tan (9 = and tan0 = du At the instant when x passes through the value a, u and v being zero by the hypothesis, P passes through the origin ; the corresponding value of 6 is evidently determined by the direc- tion in which P is moving at that instant, and is therefore equal to the value of (j) at that point. Hence the values of tan 6 and tan (j) corresponding to ;ir = ^ are equal, or therefore, to determine the value of - for ;ir = a: we substitute 21 for it the function -7— , whose value is the same as that of the du given function, when x = a. 82. This result may also be expressed in the following man- ner : let f(x) and (l>{x) be two functions, such that f{d) = o, and (l>(a) = o ; then {a) \d) (I) As an illustration, let us take log^ When x=i, this func- tion takes the form — ; by the above process, we have 82 EVALUATION OF INDETERMINATE FORMS. [Art. 82. log^n ^n ^-iJi I Ji the required value. dv fix) 83. Since the substituted function -^ or -^-rr—i frequently du (p {x) takes the indeterminate form, several repetitions of the process are sometimes requisite before the value of the function can be ascertained. For example, the function — takes the form - when tf o 6 = o\ employing the process for evaluating, we have — cos _ sin 8~\ v/hich is likewise indeterminate ; but, by repeating the process, we obtain - cos ^"1 _ sin ff cos 6~ = h 84. If the given function, or any of the substituted func- tions, contains a factor which does not take the indeterminate form, this factor may be evaluated at once, as in the following example. The function (l — ;ir) f^ — I tan'' X is indeterminate for x = O. By employing the usual process once, we obtain (l - X)8^ — l "| _ —X6^ "I tan' X J o~ 2 sec';ir tan xj J which is likewise indeterminate ; but, before repeating the pro- cess, we may evaluate the factor -^— . The value of 2 sec X |q this factor is — J ; hence we write § XiL] ABBREVIATED METHODS. 83 a=- (l — ;ir) f-^ — i-| _ _ x^^ tan' X sec' X 85. When the given function can be decomposed into fac- tors each of which takes the indeterminate form, these factors may be evaluated separately. Thus, if the given function be (f-^ — i) tan';t: may be employed. We have tan X I = I, and — !■ I = I ; -Jo X _lo hence the value of the given function is unity. When this method is used, if one of the factors is found to take the value zero while another is infinite, their product, being of the form o • 00, must be treated by the usual method, since o • 00 is itself an illusory form. 86. Another mode of decomposing a given function is that of separating it into ^^arts, and substituting the values of such parts as are found on evaluation to be finite. As an illustration, we take the expression, _ (g-^ - 8-^y- 2x\e^ + 8-^) 1- Each of the fractions into which this function can be decom- posed being obviously infinite, we 'first apply the usual process, thus obtaining 84 EVALUATION OF INDETERMINATE FORMS. [Art. Z6, fi,Q — — - ^- 4^ Separating this expression into two fractions, thus, — (f-r 4. ^-r) (f.r __ g-r _ 2x)~\ 6^ — g-^ n Jo 2X Jo ' Uo = 2X^ the latter is found on evaluation to have a finite value, and the expression reduces to e^ — 8-^ — 2x~] "° = — p — J„- '• Hence "» == — IP— 1- ' = -6^1- ' = - *• Examples XI L / ^ sin x~\ tan ^"1 , f' — i"l V I. Prove = I, = I, and = X _Jo X |o X |o I. These results are frequently useful in evaluating other functions. Evaluate the following functions : V 6* — f 2. -. , , when x^^ o. 2. log(i+^) / or - x"" log ^ — log a; J ■y'-5-^'+ 7-^-3 4- ^»_>_5^_3 » -^-3. ^ / X* — d>x^ + 22:1:^ — 24JC + 9 _ I ^* ^* — 4^" — 2Jt:' + 12^ + 9 ' -^ — 3- - / « £f-r > 6**— I :r = o. — I . § XII.] EXAMPLES. 85 f sin ^ — cos X is. , when X = In, I V2. sin 2X — cos 2X — I log JC V(i — X) o. s/ 9- -^— » ^ = °- log-. V 10. — ^ —-^ -, (See Art. 84), x = 1. — . \r V „. ?i^"-(x - cos ^), X ^ o. V' 12. , x^=a. wf" .;c — ^ / 13. :j -. , r. — \7t. aloga. / I — COSX' I V 14- ~i 7 r> .r; = o. — . * ^ X\0%{\ -\- XY 2 i/:r tan x 15. -^, x=o. I. /•?// /« the form i/ • . See Art. 2>< and Ex- ample i. Vx — Va -V- |/(-^ —a) _ I JC^/C^JC — 2JC*) — X^ 81 I 17. ^^^ r y x=i, — ^ ' 1-x^ .20 / ^ (d" + ax^- x"")^ - (a" - ax ■\- x^)^ ^/ 18.^ , \ -T ^, x = o. Va {a + xy^ — {a — xY Multiply both terms by the two complementary surds. See Art. 80. 86 EVALUATION OF INDETERMINATE FORMS. [Ex. XI I. • \\ 10. -^ -, ^^ n , when x-=-a. — ; -r . Divide both terms by (a — x)^. smx — X cos ^ ^ 20. V 24. X — smx * «* - €~' - 2X Jt^— tanjc ' {x- 2)e' + X -\- 2 xia'-iy x" — X I — X -{- log X* tan x — smx Sin X 1 sec ^ "~~ i I jPut in the form • 5 . ^ Jo X _\o J (^ — i)^ + sin^(jc'' — i)^ ^^' (^ + i) (^ - i)* ' 28. ^TT — tan~' ^ ^n ^8in(logx) f X = O. 2. ^= O. V/ 21. ;v = o. ^=1. '^2. I — V(2X — X) t sin^jc — log(6''cos^) yj 27. -^ , I a: = 0. 2' I X — I. 2(l-«)- x = o. (I 4-d!')sec'«. i tan (^ + -y) ~ tan (d? — x) V ^9' tan-' (a + x) — tan"' (a - ^)' J xsinx — irr ^ — 1 ^ >/ -o. , x = i7r. —I. ^ cos X § XII.] EXAMPLES. 87 fx c 8in a \i %\. -. — , when ^ = o. i. nf sin nx — if sin mx J -12. , m = n. ^ ^ tan ?tx — tan mx if~^{n cos nx — sin nx^ cos'' nx. Ifi solving this and the follcnmng example, x and n inay be regarded as co7istants, a?id m as a variable. ytan nx — tan mx sec" nx ^^ sm {nx — mx) 2« XIII. TAe Form -^. f(x) 87. Let ^;^ denote a function which assumes the form (p(x) -^ when x=. a, then we have I (I) The second member of this equation takes the form - when o x = a\ we therefore have, by equation (i) Art. 82, I ^'(^) /w_ .-K^) 4>{x) I f{a) _ ({(^) _ [(^(^)]' _ \a) \Ad) y whence, if •^; / is neither zero nor infinity, we infer that (pia) 88 EVALUATION OF INDETERMINATE FORMS. [Art. 87. {a)- 4>' ia) ^^^ This formula, it will be observed, is identical with that employed when the function takes the form — . o 88. When the value of "44-^ is either zero or" infinity, equa- tion (2), Art. 87, will be satisfied independently of the exist- ence of equation (3) ; we are not justified therefore, when this is the case, in deriving the latter from the former. The follow- ing demonstration shows, however, that equation (3) holds in these cases also. First, when the value of 4; x ^^ zero, by adding a finite quantity n to the given function, we have a function which is by hypothesis finite. To this function there- fore the demonstration given in Art. 87 applies ; hence therefore -^ =-^ tneretore ^^^^ ^.^^y as before. Again, if the value of v^-t is infinite, that of ^^r~ is zero, {a) /(a) and, by the last result, 4 {a) _ 'ia) f{a)-f{a)' hence, in this case, likewise § XIII.] THE FORM f . 89 f{d)_f\a) Derivatives of Functions which assume an Infinite Value, 89. When f(x) becomes infinite, for a finite value a of the in- dependent variable, f '(a) is likewise infinite. For, let b denote a value of X so taken that fix) shall be finite iov x =^ b and for all values of x between b and a : then, as x varies from b to a, the rate of /(;ir) must assume an infinite value, otherwise /"(jtr) would remain finite. The value of x for which the rate is infinite must be a or some value of x between b and a ; that is, some value of X nearer to a than b is. Now, since b may be taken as near as we please to a, the value of x for which the rate is infinite dx cannot differ from a. The expression for this rate is/"(4r) — -, in which -^ may be assumed finite, therefore f'{x) must be infinite when X = a; in other words, f'(a) is infinite when f(a) is infinite. 90. It follows from the theorem proved in the preceding article that when a is finite the function obtained by the appli- cation of formula (3), Art. 87, takes the same form, — ,as that 00 assumed by the original function. Hence, except when the given value of x is infinite, the application of some other process, either to the original function or to one of the substituted func- tions, is always requisite. Thus in the example, log (sin 2xy^ _ 00 ^ log sin ;r Jo 00 * by using the above formula we obtain 90 EVALUATION OF INDETERMINATE FORMS. [Art. 90. log sin 2x~\ _ 2 cot 2x' G log sin ^ Jo cot X 00 which takes the form — ; but the last expression is equivalent 00 sin X cos "Zx I to 2 — , and is therefore easily shown to have the sm 2x cos xAo value unity. The Form o . oo. 91. A function which takes this form may, by introducing the reciprocal of one of the factors, be so transformed as to take either of the forms - or _ , as may be found most convenient. o 00 For example, let us take the function which assumes the above form when x = ^^n being positive. In this case it is necessary to reduce to the form — . Thus — x-n^ — -^\ = = _ ^ , etc. By continuing this process, we finally obtain a fraction whose denominator is finite while its numerator is still infinite. Hence we have, for all finite values of n, X-'^ £^1 = 00. J 00 The Form oo — oo. 92. A function which assumes this form may be so trans- formed as to take the form - . Let the given function be § XIII.] THE FORM 00-00. Ql r J l og (I +-^) "] which takes the form oo—(x, since the second term is easily- shown to be infinite. But r I _ log (I +-^) 1 _ ;ir~(i +.y)log(l 4-.r) "] ^ " Lr(i + X) x' Jo ;rXi + x) Jo _ ;r-(l +;r)log(l + .y)" ! Jo 2 ;i;' I — log (i 4- x) — i 2X Examples XIII. Evaluate the following functions : sec^ sec ^x a' \l 2.- cosec (ma ') * V ^. i^^ when X = ^zr. — 3. X = CO. m. {n > o), x=. 00. /. tan jc log {x — \7t) ' / sec (JTTjc) ^ 5- log(i_^)' / g log cos (iTT jy) ^ • log(i-^) ' X = J;r. 00. .ijf = I. 00. JP = I. I. 92 EVALUATION OF INDETERMINATE FORMS. [Ex. XIII. tan^ log(i +:r) / <>• ^^ » JC = 00. o. ^ 9- (^^' — ly^, a: =^00. log ^. TtX >! X I 13. £ -(l — logx), log tan Jt ' log cot — JIC = o. COtJt' + log^' >) 17. SQCx{xsmx — ^rc), x = ^7i:. X = a. A i8. log (2 - - ) tan — , \) 19. (i — ^) tan (^7rx)y x = 1 J 20. log {x — a) tan (^ — a). X =a. y 4 V 10. ^ — tan — , X = a II. x'*{\ogxY\ (m and Xi being positive)^ x = o. o. ^ 12. £^sin-, ^=00. 00. / TtX ^ 1 2 V 14. sec log-, x=i. — . 2 ° X Tt J logtan;zx V 15. _^_ ^ ^ = 0. I. — I. § XIV.] THE FORMS c»% o\ AND i °°. 93 XIV. Functions whose Logarithms take the Form oo . o. 93. In the case of a function of the form u"^ we have log uy=v log u. The expression v log u takes the illusory form o • oo in two cases : first, when v — o and log u — on \ and secondly, when v= CO and log u — O. Log ?/ is infinite when u^o, and also when u= co] there- fore the first case will arise when the original function takes one of the forms 00° or 0°. 'Logu = o when u= i, therefore the second case will arise when the original function takes the form i °°. Hence functions which take either of the three illusory forms. 00°, 0°, or 1% may be evaluated by first evaluating their logarithms, which take the form o • co. It is to be noticed however that 0°° and 00°° are not illu- sory forms, since their logarithms take the form 00 (ip 00). The Forfn i °". 94. As an illustration of this form, we take the function which assumes the form i °° when ;ir = 00. Denot- ing this function by u, we have (-1) log2^ = ;irlog/^l -h -j the last expression assuming the form - when x — (j^. 94 EVALUATION OF INDETERMINATE FORMS. [Art. 94. In evaluating this logarithm, it is convenient to substitute z ior — \ then , log(i4-^^)"l since, when ;tr = 00, z — o. Taking derivatives, we have Z Jo I + ^^Jo Hence u^=\\-\--\ = a*. 95. If <2: = I, we have that is, as x increases indefinitely, the limiting value of the func- tion ( I H — j is f. The Napierian base is often defined as the limiting value of this function, or, what is the same thing, by- formula f=(i ^x)i\^. The Form o°. 96. The function jtr^j^, by the aid of which many functions of similar form may be evaluated, will serve as an illustration of the form 0°. Let u = x^\ then logu = x\ogx, § XIV.] THE FORM o\ 95 and logu] = -^-^ = — -^ = o; _Jo -^ _lo -^ -Jo therefore x^\ I. The value of a function which takes the form o° is usually found, as in the above example, to be unity. This is not, how- ever, universally true, as the function a + x (one of those earliest adduced for this purpose *) will show. This function takes the form o°, when x — 0\ but since its logarithm reduces to a -\- x, its value when ;r = o is £^. Examples XIV. >/ I. (cos ^)'=°^'-^, when;t: = o. e~* J /tan^Xja / -- ^ 3. (cosa'Jtr)^oseca/j.r^ ^ = O. 6^^'. — 5. (tan^m :^=-i7r. i. £, nI 6. ('^„')"(^>o). Kx"" o. I. vl 7. (i-^)% * = o. ^. 4 8. (sinjc)««^'^ ar = j7r. £-*. * See Crelles Journal, vol. xii, p. 293. 96 EVALUATION OF INDETERMINATE FORMS, [Ex. XIV. Solution: (cot ^r^]^= [^-|&° = i. (5.. ^r/. 96.) ^' 10. (sin xf^', x = o. .1. V 20. (i ± ^)^, /21. ;r"(sin^)*^°Y '^~^-^ V, \2 Sm 2JC/ {m' — i) (asinx — sin dJJi;) 22. _ ^ 2 ■>1 II. (sinjt:)'^"', a v/l2. ^log^i^, ^ 13. (sinj^y^et^-^, ^=0. e'**. x = o. V 14. ^^ (^ > o), V 15. (^')log(-^ + logcosjr)^ V — 16. :x:^ -^j when ^ = i. V 17. •^'"S :^ = o. I. X = o. f2«'. .^t: = o. ^ 18. (cos ^zj*;)-^", ^ = 0. f-i«^2 /:..('^)t ;r =: 00. .:r = 00. n X = — . 2 ,m+3 ^-^ sin ^ (cos X - cos ^^)» ' "" - °' ( 3 ] ^^S"^- V / CHAPTER VI. Maxima and Minima of Functions of a Single Variable. XV. Conditions Indicating the Existence of Maxima and Minima, 97. If, while the independent variable increases continu- ously> a function dependent on it increases up to a certain value, and then decreases, this value of the function is said to be a maximum value. In other words, a function f{x) has a maximum value corresponding to ;r = ^, if, when x increases through the value ^, the function changes from an increasing to a decreasing function. Since f'{x) is positive, when f{x) is an increasing function, and negative when it is a decreasing function ; it is obvious that li f(d) is a maximum value of f{x)yf'{x) must change sigUj from + to — , as ;r increases through the value a. On the other hand, a function is said to have a minhnum value for x= a^ if it is a decreasing function before x reaches this value and an increasing one afterward. In this case, f'(x) changes sign from — to +. 98. The derivative f'{x) can only change sign on passing through zero or infinity. Hence a value of x, for which f{x) is a maximum or a minimum^ must satisfy one of the two follow- ing equations : f\x) = o and /'{x) = oo. 98 MAXIMA AND MINIMA. [Art. 98. The required values of x will therefore be found among the roots of these equations. The case which usually presents itself, and which will there- fore be considered first, is that in which the required value of ;r is a root of the equation f\x) = o. 99. As an illustration, let it be required to divide a number into two such parts that the square of one part multiplied by the cube of the other shall give the greatest possible product. Denote the given number by ^, and the part to be squared by X ; then we have f(x) = x\a-x)\ It is evident that a maximum value of this function exists ; for when ;ir = o its value is zero, and when x = a its value is again zero, while for intermediate values of x it is positive ; hence the function must change from an increasing to a decreas- ing function at least once, while x passes from the value zero to the value a. Taking the derivative of this function, the equation is in this case 2x{a — xy — 34:^^ {a — x^ = o, or x{a — xy {2a — $x) = o. o and a are roots of this equation ; but, as we are in search of a value of the function corresponding to an intermediate value of Xy we put 2a — ^x = o, and obtain x = ^a. The corresponding value of the function is -^^a^y the maxi- mum value sought. §XV.] GEO ME TRIG A L MA GNI TUBES. 99 Maxima and Minima of Geometrical Magnitudes, 100. When the maximum or minimum value of a geometri- cal magnitude limited by certain conditions is required, it is necessary to obtain an expression for the magnitude in terms of a single unknown quantity, such that the determination of the value of this quantity will constitute the solution of the prob- lem. For example : let it be required to deterinine the cone of greatest convex surface among those which can be inscribed in a sphere whose radius is a. Any point A of the surface of the sphere being taken as the apex of the cone, let the diagram represent a great circle of the sphere passing through the fixed point A, If we refer the position of the point P to rectangular coordinates, and take C as the origin, the required cone will evidently be determined when X is determined. We have now to express the convex surface vS in terms of x. The expression for the convex surface of a cone gives S=ny^/lf^{a^xy\ ..... (l) in which the unknown quantities x and y are connected by the equation of the circle X-'^f^d^ (2) Substituting the value of j/, we have 5 = TT 4/(^' - ^0 4/(2^' + 2ax\ reducing, S=7t V{2a) {a + x)V{a—x) (3) Fig. 10. lOO MAXIMA AND MINIMA. [Art. lOO. Since the factor n V(2a) is constant, we are evidently re- quired to find the value of x for which the function /{x) =z{a -i- x) Via - x) is a maximum. The equation /\x) = o is, in this case, a -\- X V{a - x) whence 2 V{a — x) x = \a. = o; The altitude of the required cone is therefore \a. Substi- tuting this value of x in equation (3), we have S=^Vy7ta\ the maximum value required. fOI. As a further illustration, let it be required to determine the greatest cylinder that can be in- scribed in a given segment of a pa- raboloid of revolution. Let a denote the altitude, and b the radius of the base of the seg- ment. The equation of the gener- ^f x_ ating parabola is of the form V = A^cx. Since (^, b) is a point of the curve, we have the condition b"^ = 4ca ; eliminating 4^, the equation of the curve is b' Fig II. y = - ;ir. a (I) § XV.] GEOMETRICAL MAGNITUDES. 10 1 The volume V of the cylinder of which the maximum is re- quired is expressed by V — ny'ia — x)^ or, by equation (i), V — n — x{a — x). Hence we put fix) — ax — x\ and the condition /'{x) = o gives X = ^a. Consequently a — Xy the altitude of the cylinder, is one half the altitude of the segment. Examples XV. y I. Find the sides of the largest rectangle that can be inscribed in a semicircle of radius a. The sides are a V2 and \a V2. yj 2. Determine the maximum right cone inscribed in a given sphere. The altitude is four thirds the radius of the sphere. y 3. Determine the maximum rectangle inscribed in a given segment of a parabola. The altitude of the rectangle is two thirds that of the segment. J 4. Find the maximum cone of given slant height a. The radius of the base is \a V6. \^ 5. A boatman 3 miles out at sea wishes to reach in the shortest time possible a point on the beach 5 miles from the nearest point of the shore ; he can pull at the rate of 4 miles an hour, but can walk at the rate of 5 miles an hour ; find the point at which he must land. Express the 7vhoh time hi terms of the distance of the required point from the nearest point of the shore. He must land one mile from the point to be reached. 102 MAXIMA AND MINIMA. [Ex. XV. V 6. If a square piece of sheet-lead whose side is a have a square cut out at each corner, find the side of the latter square in order that the remainder may form a vessel of maximum capacity The side of the square is \a. %/ 7. A given weight is to be raised by means of a lever weighing n pounds per linear inch, which has its fulcrum at one end, and at a fixed distance a from the point of suspension of the weight w ; find the length of the lever m order that the power required to raise the weight may be a minimum. /2a'w J . ^^ V 8. A rectangular court is to be built so as to contain a given area <:', and a wall already constructed is available for one of the sides ; find its dimensions so that the least expense may be incurred. The side parallel to the wall is double each of the others. ^ J 9. Determine the maximum cylinder inscribed in a given cone. The altitude of the cylinder is one third that of the cone. 10. Prove that the rectangle with given perimeter and maximum area is a square , also that the rectangle with given area and minimum perimeter is a square. i II. Find the side of the smallest square that can be inscribed m a square whose side is a. Take as the independent variable the distance between the angles of the two squares. ia ^2. ^ 12 Inscribe the maximum cone in a given paraboloid, the apex of the cone being at the middle point of the base of the paraboloid. The altitude of the cone is half that of the paraboloid. 13. Find the maximum cylinder that can be inscribed in a sphere whose radius is a. The altitude is ^a V3. s^ 14. Through a point whose rectangular coordinates are a and b draw a line such that the triangle formed by this line and the coordinate axes shall be a minimum. The intercepts on the axes are 2a and 2d, § XV.] EXAMPLES. 103 y / 15. A high vertical wall is to be braced by a beam which must pass over a parallel wall a feet high and b feet distant from the other , find, the length of the shortest beam that can be used for this purpose. Take as the independent variable the inclination of the beam to the horizon {J^ + P) . 16. The illumination of a plane surface by a luminous point being directly as the cosine of the angle of incidence of the rays, and in- versely as the square of its distance from the point ; find the height at which a bracket-burner must be placed, in order that a point on the floor of a room ^t the horizontal distance a from the burner may jeceive the greatest possible amount of illumination. The height is —y. XVI. Methods of Discriminating between Maxima and , Minima, 102. When the existence of a maximum or a minimum cor- responding to a particular root a of the equation f\x) = o is not obvious from the nature of the problem, it is necessary to determine whether f\x) changes sign as x passes through the value a. If a change of sign does take place we have, in accordance with Art. 97, a maximum if, when x passes through the value a, the change of sign is from -h to — ; that is, if fix) is a de- creasing function, and a minimum if the change of sign is from — to +, in which case f'{x) is an increasing function. 103. In many cases we are able to distinguish maxima from minima by examining the expression for f'{x), as in the fol- lowing examples. 104 MAXIMA AND MINIMA. [Art. IO3. Given /W = l4^' whence /'(^) = l^|£_-_i ,. f\x) — O gives log X — \y or ;ir = f. Since log;r is an increasing function, it is obvious that, as x in- creases through the value ^yf'{x) increases ; it therefore changes sign from — to +, and consequently f{e) is a minimum value of/W. 104. If f\x) does not change sign we have neither a maxi- mum nor a minimum ; thus, let f{x) = ;i; — sin;i:, whence f'{^) = i — cosjir. In this case /'{x) becomes zero when x = 2nn^ n being zero or any integer, but does not change sign, since i — cos;t- can never be negative ; consequently fix) has neither maxima nor minima values, but is an increasing function for all values of X. Alternate Maxima and Minima, 105. Let the curve be constructed, and suppose it to take the form represented in Fig. 12. There is a maximum value of f(x) at B, another at D, and minima values occur at ^, at C, and at E. It is obvious that in a continuous por- tion of the curve maxima and minima ordinates must occur alternately, and ^ must separate the curve into segments in which the ordinate is alternately an increasing and a decreasing function ; hence, if f(x) has maxi- ^ X § XVI.] ALTERNATE MAXIMA AND MINIMA. IO5 ma and minima values, they must occur alternately unless infi- nite values of the function iiitervene. It is also evident, with the same restriction, that a maximum is greater in value than either of the adjacent minima, but not necessarily greater than any other minimum ; thus, in Fig. 12, the maximum at B is greater than the minima at A and C, but not greater than that at E. 106. As an illustration let us take the following function in which it is easy to discriminate between the maxima and min- ima values. f{x) = x{x^df(x-ay. Whence, f\x)^{x + aj {x - ay + 2x{x ^ a)(x - af + ix{x + a)' {x - a)\ =={x-\-a){x- ay {6x' + ax - a'). a and — a are evidently roots of f\x) = o ; the roots derived by putting the last factor equal to zero and solving are — ^a and ^a. Hence /'(x) can be written in the form f'{x) = 6{x + a) {x + ia) {x - ia) {x - d)\ in which the factors are so arranged that the corresponding roots are in order of magnitude. When X < — a, f'{x) is negative, and, if we regard x as in- creasing continuously, f\x) changes sign when x = — a, when X = — ^a, and again when x = ^a, but not when x = a. Since /'{x) is at first negative it changes sign from — to + when it first passes through zero, that is when x = ~ a; the corresponding value of /(x) is therefore a minimum. Accord- ingly the value of /(x) corresponding to the next root x = — ia is a maximum, and that corresponding to x = ^a is another minimum ; but there is neither a maximum nor a minimum corresponding to x := a. I06 MAXIMA AND MINIMA, [Art. lO/. 107. When the function is continuous as in the above ex- ample, that is, does not become infinite for any finite value of Xy it is always easy to determine by examining the function itself whether the last, or greatest value of x in question, gives a maximum or a minimum. Thus, in the above example, f{x) evidently increases without limit as x increases without limit ; therefore, the last value must be a minimum. K The Employment of a Substituted Function, 108. Since an increasing function of a variable increases and decreases with the variable, such a function will pass from a state of increase to a state of decrease, or the reverse, simulta- neously with the variable ; that is, it will reach a maximum or a minimum value at the same time with the variable. This fact often enables us to simplify the determination of maxima and minima by substituting an increasing function of the given function for the given function itself. For example, if we have f{x) = V{b' -f ax) + V{b^ - ax\ we may with advantage employ the square of the given func- tion. The square is 2b'' ^2s/{b'-a'x% which is obviously a maximum when x — o^ and, since the square of a positive quantity is an increasing function, we infer that f(x) is likewise a maximum for the same value of x, 109. A decreasing function of the given function may also be employed ; but, in this case, since the substituted function decreases with the increase of the given function and increases v/ith its decrease, a maximum of the substituted function indi- § XVI.] SUBSTITUTED FUNCTIONS. lO/ cates a minimum, and a minimum indicates a maximum of the given function. Thus, if we have X f{x) = x' — 3x -\- i' the reciprocal may be employed. The reciprocal of this func- tion is x^ — 2,x -{- I I X X whence, taking the derivative, we obtain ^ _ ;ir2 — I *^ :? ~x~ ' which vanishes when x = ± i. Since x^ is an increasing function when x is positive, this deriv- ative is evidently an increasing function when x = i. The re- ciprocal is therefore a minimum for this value of x, and conse- quently f(\) is a maximum value of fix). In a similar manner it may be shown that /(— i) is a minimum. Examples XVI. Determine the maxima and minima of the following functions : I- f \x) = ^. .A min. for jt = - . 2. f\x) = — ^ . A max. for x = f^. 3. /(x) — — _3-^^ . A min. for x = ^a. /4./W^ /^^\^^^y * Amin. for^= - ^V- ^OS MAXIMA AND MINIMA. [Ex. XVI. */ 5- /W = sin 2x —x. A max. for x = nTr + ^tt ; a min. for x = nTt ~ ^jt. ^ 6. /(^) = 2:v' + 3x' — 2>6x + 12. A max. for .^ = - 3 ; a min. for x ^= 2. J 7. f{x) = x' — sx' — gx + S' A max. for jt; = — i ; a min. for x = z. J 8. f{x) = 3^'— i25.jt' + 2i6ojc. A max. for ;»:=:— 4 and x=:^ ; a min. for ^=—3 and .^=4, v g. f(x) ^= b + c{x — a)^. Neither a max. nor a min. yl 10. /(^) = (^ — i)* (jc + 2)'. A max. for ;<; = ~~ t 5 a min. for x = i. y II. /(x) = {x — gY {x — 8)*. A max. for jc = 8 ; a min. for x = -8f . / - . ^ I — :r + .;«?' ^ 12. fix f {oc) = — ; ^ . A min. for x = h •^ ^ ^ 1 + X — x^ ^ /^ / N ax c^ A ^ Max. for ;<: = i ; j Min. f or ^ = ~ i (^ being positive). V I5./(^) = (I+^I)(7_^)^ iSi?/z'(? by putting x =^ z^. For method of discriminating between max- ima and minima, see Art. 107. Min. for ^ = o, and ^ = 7 ; Vmax. for .x=: i. 16. f{x) — 5^1?' + 12^' — 15^ — 4o:r' + \^x^ + 60^ + 27. Min. for ^ = — 2. J 17. f{pc) = jc" — 6^* 4- 4^^ + 9^' — ^2x + 3. Min. for :r = — 2, and .^ = i; max. for .%: = — i . ^Q/ t • • § XVI.] EXAMPLES. 109 y 18. The top of a pedestal which sustains a statue a feet in height is b feet above the level of a man's eyes ; find his horizontal distance from the pedestal when the statue subtends the greatest angle. When the distance = \/\b[a + h)\. 19. It is required to construct from two circular iron plates of radius a a buoy, composed of two equal cones having a common base, which shall have the greatest possible volume. The radius of the base = \a 4/6. i/ 20. The lower corner of a leaf of a book is folded over so as just to reach the inner edge of the page ; find when the crease thus formed is a minimum. Solution : — Let J/ denote the length of the crease, x the distance of the corner from the intersection of the crease with the lower edge, and a the width of the page. By means of the relations of similar right triangles, the following expression is deduced : _ X Vx ^~ V{x-lay Whence we obtain x=%a, which gives a minimum value oi y. yy ?i. Find when the area of the part folded over is a minimum. When X ='^a. XVII. The Employment of Derivatives Higher than the First. MO. To ascertain whether /'(;tr) is an increasing or a de- creasing function, (and thence whether /(;r) is a minimum or a maximum), it is frequently necessary to find the expression for its derivative, /"(;ir). Now, \i f'\d) is found to have a positive value, it follows that f\x) is an increasing function when x — cu no MAXIMA AND MINIMA. [Art. I lO. and, as was shown in Art. 102, that/(rt) is a minimum. On the other hand, if we find \.h^tf"(a) has a negative vakie, it follows that/'(^) is a decreasing function, and that/"(^) is a maximum. To illustrate, let f{x) — ix' — iGx" — Gx" + 12, then f\x) — \2x'' — ^Zx" — \2x. The roots of fix) — o are x ~ o, and x — 2 ± VS- In this case /"{x) = i^x"" — g6x — 12, hence /"(o) = — 12 ; /{x) is therefore a maximum when ;ir = o. It is unnecessary to find the values of y"(;ir) for the other roots ; for, since the function does not admit of infinite values, the maxima and minima occur alternately. The root 2 — V S being negative and 2 + 4/5 positive, the root zero is intermediate in value, and therefore both the remaining roots give minima. [|(, If /'(x) contains a positive factor which cannot change sign, this factor may be omitted ; since we can determine whether /'{x) increases or decreases through zero by examin- ing the sign of the derivative of the remaining factor. Thus, if Since z- ^r^ is always positive, we have only to determine (i 4- ^ ) whether the factor i — x"^ changes sign. Denoting this factor by V, and putting v = o, wq have x= ±1. Now -r- = — 2X ax which is negative for ;if = i and positive for x——\. These § XVII.] EMPLOYMENT OF SECOND DERIVATIVES. Ill roots, therefore, give respectively a maximum and a minimum value o{ f{x). (12. There may be roots of the equation f'{x) — o which correspond to neither maxima nor minima, since it is a condi- tion essential to the existence of such values that f\x) shall change sign. When such cases arise, the form assumed by the curve y — f(x) in the immediate vicinity of the point at which X ^=^ a will be one of those represented at A and B in Fig. 13- At these points the value of tan ^ or f\x) is zero, but at A it is positive on both sides of the point, and fix) or y is an increasing function, while at B fix) is negative on both sides of the point, Fig. 13. and f{x) is a decreasing function. il3. It is important to notice that at A the value zero assumed by f\x) constitutes a minimum value of this function, thus a root of f'{x) — o for which /'{^) is a niinimuni corre- sponds to a case in which f{x) is an increasing function. In like manner a root of f\x) = o for which /'{x) is a maximum is a case in which f{x) is a decreasing function. H4-. It follows from the preceding article and from Art. 102 that, if/'(^) = O, then, of the two functions /(;ir) and/'(.r), one will be a maximum and the other a decreasing function, or else one will be a minimiun and the other an increasing function. Hence, if we consider the case in which the given function and several of its successive derivatives vanish for the same value of x, it is evident that when these functions are arranged in order they will be either alternately maxima and decreasing functions^ or alternately minima and increasing func- tions. lis. Now suppose that ^{x) is the first of these successive 112 MAXIMA AND MINIMA. [Art. 1 1 5. derivatives that does not vanish when x = a^ then, writing the series of functions /W, /'W. /"W. /'-"W, /V), let us assume first that f"{a) is positive. Then in the above series of functions f"~\ci), f~\<^\ ^tc, will be increasing functions while /"~X<^), f"~\a), etc., will be minima. Now whenever 7t is odd, the original function will belong to the first of these classes and will be an increasing function, while if n is even the original function will belong to the second class and will be a minimum. On the other hand, if f'\d) has a negative value, the series of functions will be alternately decreasing functions and maxi- ma ; and when n is odd f{a) will be a decreasing function, but when n is even f{d) will be a maximum. Thus we shall have neither maxima nor minima unless the first derivative, which does not vanish when x — a, is of an even order ; but when this is the case we shall have a maximum or a minimum according as the value of this derivative is nega- tive or positive. 116. The following function presents a case in which the above principle is advantageously employed. f{x) = £-^ + £~"^ + 2 cos X, f'{x) = £^ — e~^ — 2 sin x. Zero is evidently a root of the equation /'{x) — o.* In this case * Zero is the only root of /'{x) = o in this example ; for / (x) = > ' . f"{x) therefore cannot be negative, hence f'{x) cannot again assume the value zero. § XVII.] INFINITE VALUES OF THE DERIVATIVE. II3 f"{x) = f"^ + f""^ — 2 cos;ir .'. /"(o) = O, f"\x) = €^ — a~^ 4- 2 sin ;ir .-. /'"(o) = O, /''(x) = 6^ -{■ €~^ + 2 COS ;ir .-. /'^ (o) = 4. The fourth derivative being the first that does not vanish, and having a positive value, we conclude that x = o gives a mini- mum value of /{x). Infinite Values of the Derivative, (17. It was shown in Art. 98 that if we have, for x = a^ f\x) = «, a maximum value will present itself \i f\x) changes sign from + to — , and a minimum if it changes sign from — to +. It may, however, happen in these cases that the value of f (a) is also infinite. When/(^) is finite, the form of the curve in the vicinity of a maximum or a minimum ordinate of this variety is represented at A and B in Fig. 14. As an example, let whence /(;r)=(;r* - /^i)* f\x) = %x'\x^-lyf)'^. k fix) is infinite when x =0 and when x = b. When X = o fix) does not change sign, since x~^ cannot be negative, but when x = b q it changes sign from — to + ; hence fix) has a minimum value when x =. b. X Fig. 14. \/ 114 MAXIMA AND MINIMA. [Ex. XVII. Examples XVII. y I. Show that ae^' + bs~^' has a minimum value equal to 2 V{(ib). Find the maxima and minima of the following functions : 2. f(x) = X sin X. A maximum for a value of x in the second quadr^int satisfying the equation tan x ^= — x. J .. . a^ b^ 3- /(^) = - + - X a The roots x = r and x = r give a min. and a max. if b is « + /^ a — b^ positive, but a max. and min. if b is negative. ^ 4. /(jc) = 2 cos X + sin'^ .r. Solution : — f {.^^ =^ 2 sin jjc (cos^ — i) J rejecting the factor 2(1 — cos x\ which is always positive, we put V — — sm X. Hence -^ = — cos x. y A max. for .;c = 2n7t ; a min. for x = (2;? + i) rr. 5. /(jc) = sin x{i -\- cos ^). A max. for ^ = Jtt ; a min. for x=^ — \7t ; neither for x = 7t, j 6. f {x) = sec X + log cos'' x. Multiplying the derivative by cos^ x^ we obtain ,^ . z; = sinj[:(i — 2 cos^i;). c^ A A max. for ^ = o, and x-=- n \ n-ftw , ^ -H^ < a min. for ^ = ± J tt. xf \ __ tan' ^ A min. for ^ = o, |7r, |;r, and n ; ' ""^ ^ tan 3JIC * a max. for Jt: = ^tt, \7t^ ^tt, etc. J 8. /(x) = £" + f-* — .rl A min. for x = o. § XVII.] EXAMPLES. 115 / : 9. Find maxima and minima of the following functions : f {pc) = {x^ — I^) ^. A min. for ^ = o. / 10. f{pc) = {x- — l>^)-^. A max. iox x = o \ a min. for x = ^l l^* II. f{x) — {pc^ + 3^ + 2)^ + ^\ f\x) = 00 gives min. corresponding to ^= — 2, x— — i and jc=o. f'{x) = o gives two intermediate maxima. ^12. /{x) = {x^ + 2^)^ — {x + 3)^. Max. for x=i{—^± 4/17) ; min. for x ^= o and a= — 2. I / 13. /(:*:) = (^ — ^)5 (^x — ^)^ + ^. A max. for x — ; min. for x =^ a and x ^=^ b. / 14. /(^) = {x-a){x-b) A mm. for x — a + I IS. f{x) = {x- a)i (x - b)\ Solutions iox X = a and x = l{2b + a) ; if b > a, the former gives a max. and the latter a min. t Miscellaneous Examples. Max. for ^ = 4. J^/ X -^'^ — Jc + I A max. for ^ = o ; ' x^ + X — I a mm. for .v — 2. Il6 MAXIMA AND MINIMA. [Ex. XVII. 7 ~~ 3- f{^) = ■^'"" ^**- A min. for jc = ^ . / 4. The equation of the path of a projectile being y = X tan a 4^ cos"^<^ ' find the value of x when j^^ is a maximum ; also the maximum value of y. Max. when x ■=^ h'ivci2a^ and y = h sin^ a. \ 5. In a given sphere inscribe the greatest rectangular parallel- epiped. Solution : — Regarding any one edge as of fixed length, it is easy to show that the other two edges are equal. Hence the three edges are equal. ^ 6. In a given cone inscribe the greatest rectangular parallelo- piped. Solution : — Regarding the parallelopiped as inscribed in a cylinder which is itself inscribed in the cone, the base is evidently a square, and the altitude is that of the maximum cylinder. See Ex. XV, 9. V 7. A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, required the height and breadth of the window when the quantity of light admitted is a maximum. The radius of the semicircle is equal to the height of the rectangle. >J 8. A tinsmith was ordered to make an open cylindrical vessel of given volume, which should be as hght as possible ; find the ratio be- tween the height and the radius of the base. The height equals the radius of the base. ^ 9. What should be the ratio between the diameter of the base and the height of cylindrical fruit-cans in order that the amount of tin used in constructing them may be the least possible ? The height should equal the diameter of the base. § XVII.] MISCELLANEOUS EXAMPLES. 11/ V lo. Determine the circle having its centre on the circumference of a given circle so that the arc included in the given circle shall be a maximum. A max. for the value of which is in the first quadrant. /„ /x Given the vertical angle of a triangle and its area ; find when its base is a minimum. The triangle is isosceles. \/i2. Prove that, of all circular sectors of the same perimeter, the sector of greatest area is that in which the circular arc is double the radius. 13. Find the minimum isosceles triangle circumscribed about a par- abolic segment. The altitude of the triangle is four-thirds the altitude of the seg- ment. 1/14. Find the least isosceles triangle that can be described about a given ellipse, having its base parallel to the major axis. The height is three times the minor semi-axis. 15. Inscribe the greatest parabolic segment in a given isosceles triangle. The altitude of the segment is three-fourths that of the triangle. 16. A steamer whose speed is 8 knots per hour and course due north sights another steamer directly ahead, whose speed is 10 knots, and whose course is due west. What must be the course of the first steamer to cross the track of the second at the least possible distance from her ? N. 53° 8' W, 17. Determine the angle which a rudder makes with the keel of a ship when its turning effect is the greatest possible. Solution : — Let ^ denote the angle between the rudder and the prolongation of the keel of the ship ; then if b is the area of the rudder that of the stream of water intercepted will be /^ sin ^ : the resulting force being decomposed, the component perpendicular to the rudder contains the factor sin'' ^. Again decomposing this force, and taking the compo- nent that is perpendicular to the keel of the ship, which is the only Il8 MAXIMA AND MINIMA. [Ex. XVII. part of the original force that is effective in turnii.g cAc ship, the ex- pression to be made a maximum is sin" ^ cos ^. Whence we obtahi tan ^ — |/2. 1 8. The work of driving a steamer through the water being propor- tional to the cube of her speed, find her most economical rate per hour against a current running a knots per hour. Solution : — Let z; denote the speed of the steamer in knots per hour. The work per hour will then be denoted by kv^, k being a constant, and the actual distance the steamer advances per hour hy v ^ a. The work per knot made good is therefore expressed by Whence we obtain the result X^i^ h^ CHAPTER VII. The Development of Functions in Series. XVIII. The Nature of an Infinite Series, 1(8. A FUNCTION which can be expressed by means of a limited number of integral terms, involving powers of the inde- pendent variable with positive integral exponents only, is called a rational integral function. When f(x) is not a rational integral function, it is usually possible to derive an unlimited series of terms rational and in- tegral with respect to x, which may be regarded as an algebraic equivalent for the function. The process of deriving this series is called the development of the function into an infinite series. When the given function is in the form of a rational frac- tion, the ordinary process of division (the dividend and divisor being arranged according to ascending powers of x) suffices to effect the development. Thus — -i-i-^ = I 4- 2;ir + 2;ir' + 2;tr' + • • • , a series of terms arranged according to ascending powers of Xy each coefficient after the absolute term being 2. It is to be observed, in the first place, that, owing to the indefinite number of terms in the second member, the equa- tion as written above cannot be verified numerically for an assumed value of x» In this case, however, the process not I20 THE DEVELOPMENT OF FUNCTIONS [Art. 1 1 8. only gives us the series, but the remainder after any number of terms. Thus carrying the quotient to the term containing x^, and writing the remainder, we have — \ -^ 2X -^ 2X'^ ' ' ' -^ 2X'' -\ . 1 -- X This equation may now be verified numerically for any assumed value of X ; or algebraically by multiplying each member by 1 — X, thus obtaining an identity. The ordinary process of extracting the square root of a polynomial furnishes an example of a series which may be ex- tended so as to include as many terms as we please ; but this process gives us no expression for the remainder. 119. Assuming that /(x) admits of development into a series involving ascending powers of x, and denoting the re- mainder after 7i -\- i terms by R, we may write f{x) =A + Bx+ Cx'+ . . . + Nx^' + i?, . . . (i) in which A, By C, ... N denote coefficients independent of Xy and as yet unknown ; the value of R is however not indepen- dent of X. If the coefficients By C, . . . N admit of finite values, it may be assumed that i? is a function of x which van- ishes when X = O'y and in accordance wifh this assumption equation (i) becomes, when x — o, /iO) = Ay (2) which determines the first term of the series. If in any case the value of /(o) is found to be infinite, we infer that the pro- posed development is impossible. 120. When the coefficients B, C, , . . N admit of finite values, and the value of the function to be developed remains % XVIII.] XN FINITE SERIES. C^2I finite, R will have a finite value. If moreover the value of R decreases as n increases, and can be made as small as we please, by sufficiently increasing n^ the series is said to be convergent, and may be employed in finding an approximate value of the function f{x) ; the closeness of the approximation increasing with the number of terms used. A series in which R does not decrease as n increases is said to be divergent. When the successive terms of a series decrease it does not necessarily follow that the series is convergent ; for the value of the equivalent function, and consequently that of R^ may be infinite. To illustrate, if we put x — \ m the series X + J^'^ ^\x' ^\x' ^' . . , we obtain the numerical series it can be shown that, by taking a sufficient number of terms, the sum of this series may be made to exceed any finite limit, the value of the equivalent or generating function of the above series being in fact infinite when x = i.* 121. Since R vanishes with x, every series for which finite coefficients can be determined is convergent for certain small values of x. In som-e cases there are limiting values of x, both positive and negative, within which the series is convergent, while for values of x without these limits the series is diver- gent. These values of x are called the limits of convergence. * If we consider the first two terms separately, and regard the other terms as arranged in groups of two, four, eight, sixteen, etc., the groups will end with the terms \, \, -j^g, 3^, etc. The sum of the fractions in the first group exceeds f or ^, the sum of those in the second exceeds \ or \, and so on ; hence the sum of 2iV such groups exceeds the number N, and N may be taken as large as we choose. The generating function in this case is log , and unity is the limit of con- vergence. 122 THE DEVELOPMENT OF FUNCTIONS. [Art. 121. We shall now demonstrate a theorem by which a function in the form f{xo + h) may be developed into a series involving powers of h^ and in Section XIX we shall show how this theorem is transformed so as to give the expansion of f{x) in powers of x. Taylor^s Theorem. (22. A function of h of the form f{xo -\- Jt) in general admits of development in a series involving ascending powers of h. We therefore assume /(;ro + >^) = ^o 4- BJt + a>^'+ • . . + NoU" + 7?o, . . (i) in which Aoy Boj Co, . . . iVo are independent of /i, while Ro is a function of /i which vanishes when /i is zero. Hence, mak- ing ^ = o, we have /(^o) = Ao. We have now to find the values oi Bo, Co, - - - No, which are evidently functions of Xo. For this purpose we put ^i = Xo-\- hj whence h ^^ x^— Xo ; substituting, equation (i) takes the form f{x;)^f{Xo)^Bo{x,-Xo)^Co{x,-Xoy ' • • -{-JVoix.-Xof-hRoy in which we may regard Xj, as constant and Xo as variable. Re- placing the latter by x^ and its functions, Bo, Cc, . . » iVo, and Rn, by B, C, . . . Ny and R, we have f{x:)=f{x)-^B{x.^x) + C{x.-'xy-'^^N{x.^xT^R, . (2) Taking derivatives with respect to x, we have § XVIII.] TAYLOR'S THEOREM. 1 23 - nN{x. - x)-^ + {X, - xy^ + g . . . . (3) To render the development possible, B, Cj . . . JV, and R must have such values as will make equation (3) identical, that is, true for all values of x, 123 It is evident that B may be so taken as to cause the first two terms of equation (3) to vanish, and that, this being done, C can be so determined as to cause the coefficient of (>! — x) to vanish, D so as to make the coefficient of {x^ — xj vanish, and so on. The requisite conditions are f{x)-B=o, g-2C=o, g-3Z) = o,etc., A a u I .ndN dR and finally ix^ — x) — — f- -7- = o. ' dx dx From these conditions we derive B=f{x\ C=i^=i/"{x), and in general N— /"(x). Putting Xo for x, and substituting in equation (i) the values of Aoy Boy Coi . . . No, We obtain /(^„ + A)=/(;r„)+/'(^o)/^+/"(^o)-^ - . .+/«(;r„)— ^- +je„.(4) 124 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 2 3. This result is called Taylor's Theorem, from the name of its dis- coverer, Dr. Brook Taylor, who first published it in 171 5. It is evident from equation (4) that the proposed expansion is impossible when the given function or any of its derived func- tions is infinite for the value x^. / Lagrange s Expression for the Remainder, 124. i? denotes a function of x which takes the value ^o when X — Xof and becomes zero when x — x^. It has been shown in the preceding article that R must also satisfy the equation , .^dN dR or, substituting the value of N determined above, ^ = - (-^^^^/-(^) (5) This equation shows that — cannot become infinite for any ax value of X between x^ and x^^ provided /''"^'(;tr) remains finite and real while x varies between these limits. Since it follows from the theorem proved in Art. 104 that all preceding deriva- tives must be likewise finite, the above hypothesis is equivalent to the assumption that/(;ir) and its successive derivatives to the (n -f \)th inclusive remain finite and real while x varies from Xo /^ Xc + h. (25. Let P denote any assumed function of x which, like R^ takes the value Ro when x — Xo and the value zero when X = x^, and whose derivative —r- does not become infinite or dx imaginary for any value of x between these limits. § XVIIL] EXPRESSIONS FOR THE REMAINDER. 1 25 Then, Ro being assumed to be finite^ P — R denotes a func- tion of X which vanishes both when x — Xo and when x — Xt. and whose derivative cannot become infinite for any interme- diate value of X, It follows therefore that the value of this function cannot become infinite for any intermediate value of x. Since, as x varies from x^ to x^^ P — R starts from the value zero and returns to zero again, without passing through infinity, its numerical value must pass through a maximum ; hence its derivative cannot retain the same sign throughout, and as it can- not become infinite it must necessarily become zero for some intermediate value of x. Since x-, =Xo + /i this intermediate value of X can be expressed by Xo + O/i^ being 2. positive proper fraction. It is therefore evident that at least one value of x of the form x = Xo-^ Qh will satisfy the equation dP dR ,.. 'dx-^x-"" ^^) 126. The value of P will fulfil the required conditions if we assume _{Xj—Xf^ 'R. o» for this function takes the value ^o when x = Xo and vanishes when X =^ x^\ moreover its derivative with reference to ;r, viz., dP _ (« + iH-r,--i-y' Tx- li^^ ^°' • • • • w does not become infinite for any intermediate value of x. Sub- stituting in equation (6) the values of the derivatives given in equations (5) and (7), and solving for R^^ we obtain Ro=^ ^^^.~—-,r^\xo^- eh). ... (8) 1.2- • •^2.(« + l) ^ 126 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 26. This expression for the remainder was first given by La- grange. The series may now be written thus : f(x.^ h) =f{x.) +f'{x:)h +f\Xo) -^ • ■ • It should be noticed that the above expression for the remain- der after n + i terms differs from the next, or {?i + 2jth term of the series, simply by the addition of 6h to Xo^ The Binomial Theorem, 127. We shall now apply Taylor's Theorem to the function (a + by in order to obtain a series involving ascending powers Qib, In this case b takes the place of h, and a that of Xo ; hence f{x)^x /. f{x^^Xo . =a \x) = mx .,'. /\Xo) = 'mXo —ma f'\x) = m{m — \)x'"'''' .'. /"{Xo)=m(m — i)x'"~^ = m{m— i)a"~' and /"{xo) = m{m — i){m — 2) - . - (m - n + i)a'"-''. Whence (a + by"= a'"+ md'^-'b -f- '"^^ILZ^ a!"-'b' m{in - \){in - 2) . ♦ . (in - n ■\- i )^^^..^« 4. . , . 1.2.3 ' ' ' f^ This result is called the Binomial Theorem. J § XVIII.] EXAMPLES. 127 ^. Examples XVIII. To expand log {x^+ h) by Taylor's Theorem. Solution : — f(x) - log X :. /{x^} = log x^ /» = -^ .-. /■V„) = -i5 /'»=^ .-. /"'(^») = -^ /"W=-^ ••• /^Vo)=-'-^ By substituting in equation (4), Art. 123, we obtain log(^,+ ^)=log^,+ ^-— +_------... -(_i)» 4-^0. ^o ^^o Z^o 4-^0 ^^-^o Employing Lagrange's expression for the remainder (Art. 126) we derive ^ 2. Expand a=^° + \ {n + i)(^o+ ^^T'''' Solution : — 128 THE DEVELOPMENT OF FUNCTIONS. [Ex. XVIII. f{x) = a' fix) = log a-a' /'(^o) ^ log ^'^"^ f\x) = (log aY-a' .-. /»(^J - (log ay-a'^ Substituting in equation (4), Art. 123, we have ^..» == ,.. [, + log a-/> + (log aY^... + (Mflll + i?.. 1/ 3. Find the expansion of /(jc^ 4-^), when /(^) = ^ log x — x^ writ' ing the {n + i)''' term of the series. /{x^ + A) = x^ log x^—x^-h log jc„./^ + •^o 1-2 -^o *2-3 "^ ^ '^<-^ '(n- i)n 4. Expand sin"' (x^ + h) to the fourth term inclusive. ,/ ,\ • 1 h x^ h^ sm-'(-^o+ ^) - sm-^^,+ -3: + ^ • — ^ I + 2:r^ /^^ ^ (i-OV^-2-3 y] 5. Prove that j I-2-3-4-5 J ^ 6. Prove that tan (J;r + /^) = I + 2^ + 2/^= + P" + V"^' + • • • § XIX.] MACLAURIN'S THEOREM. 1 29 XIX. Maclaurin s Theorem. 128. We shall now give a particular form of Taylor's Series^ which is usually more convenient, when numerical results are to be obtained, than the general form given in the preceding sec- tion. This form of the series is obtained by putting x^=-0 and replacing h by x in equation (4), Art. 123. Thus, /(^)=/(o)+/'(o)^+/"(o) ^ ■ • • +/"(o) ^ l]^ _ ^ +R^ . . (I) and, the same substitutions bding made in equation (8), Art. 126, we obtain I -2 •••(«+ l)* Equation (i) is called Maclaurin's Theorem: it maybe used in developing any function to which Taylor's Theorem is ap- plicable, by giving a different signification to the symbol /. Thus, if log (i + /t) is to be developed by Taylor's Theorem, /{x) = log Xf the value of x^ being unity; but, if log (i -\- x) is to be developed by Maclaurin's Theorem, we must put /{x) = log (i + x), (Compare Ex. XVIII., i, with Art. 130.) The Exponential Series and the Value of e, 129. As an example of the application of the above theo- rem, we shall deduce the development of the function «-^, which is called the exponential series, and shall thence obtain a series for computing the value of f. The successive derivatives of e-^ being equal to the original function, the coefficients, y(o),/"(o), etc., each reduce to unity; 130 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 29. we therefore derive, by substituting in equation (i) and in- troducing the value of Ro, f^= I +;ir+ — + . . • + + f^-'' 2 1-2.3 1-2 n 1-2 •• . '{71 + I) Putting X equal to unity, we obtain the following series, which enables us to compute the value of the incommensurable quan- tity f to any required degree of accuracy : III f = I + I + — + ■ + • • • 1-2 12-3 I-2-34 I-2-3 • • n I-2-3 . • (;2 + i)' The computation may be arranged thus, each term being de- rived from the preceding term by division : 2.5 ,16666666667 4166666667 833333333 138888889 1984 I 270 2480159 275573 27557 2505 209 16 I 2.71828182846 Since f* is less than e, the remainder (n being 14) is less than j\ of the last term employed in the computation, and therefore cannot affect the result. Inasmuch as each term may contain a positive or negative error of one-half a unit in the last decimal § XIX.] THE VALUE OF e. 13! place, we cannot, in general, rely upon the accuracy of the last two places of decimals, in computations involving so large a number of terms. Accordingly, this computation only justifies us in writing £=2.718281828. Logarithmic Series. 130. The logarithmic series is deduced by applying Mac- laurin's Theorem to the function log(i + x). In this case /w = :log(l + X) . •• /(o) = o /w = I I + X •• /'(o)=i /"W = I •• /"(o) = -r (l+^y ■ /'"W = 1-2 ■. /'"(O) = 1-2 f\x) = I-2-3 . /-(o)=- 1.2.3, X X X hence loo^ (i -\- x) = x -H + . . . . . . (i) • ^^ ^ 234 ^ ^ Since this series is divergent for values of x greater than unity (see Art. 120), we proceed to deduce a formula for the difference of two logarithms, which may be employed in com- puting successive logarithms; that is, denoting the numbers corresponding to two logarithms by n and n + k, we derive a series for log {n ■\- h) — log n = log . 132 THE DEVELOPMENT OF FUXCTIONS. [Art. I30. A series which could be employed for this purpose might be ft ~\~ Ii h obtained from (i), by putting in the form i + -. We ob- n n tain, however, a much more rapidly converging series by the process given below. Substituting — ;ir for ;ir in (i), we have 234 log (l — ;r) = — ;ir ^ _ . . . . . (2) Subtracting (2) from (i), log = 2 ^ + ^' 3 ^^^...]. . . « a series involving only the positive terms of series (i). \ '\~ X 7t ~f" ft h Putting = , we derive x = ; substituting I X iz ^fl -\- tl in (3), v/e have The Computatio7i of Napierian Logarithfris. 131. The series given above enables us to compute Napierian logarithms. We proceed to illustrate by computing loge 10. The approximate numerical value of this logarithm could be obtained by putting n — i and // = 9 in (4) ; but, since the series thus obtained would converge very slowly, it is more convenient first to compute log 2 by means of the series obtained by put- ting n z=i I and >^ ~ i in (4) ; thus : 1 fi , I I I I I I "1 § XIX.] LOGARITHMTC SERIES. 133 We then put n= S and ^ := 2 in (4) ; whence loge 10 l0ge2 + 1+i -3 3 1^ 3^ s y 7 3""^" J* In making the computation, it is convenient first to obtain the values of the powers of ^ which occur in the series for log 2, by successive division by 9, and afterwards to derive the values of the required terms of the series by dividing these auxiliary numbers by i, 3, 5, /, etc. The same auxiliary numbers are also used in the computation of loge 10. See the arrangement of the numerical work below. 1 s 0.3333333333 I 0.3333333333 ar 370370370 3 123456790 ar 41 152263 5 3230453 ay 4572474 7 65321I ar 508053 9 56450 ar 56450 II 5132 ar 6272 13 482 ar 697 15 46 ar 77 17 5 log . 2 = 0.3465735902 2 : 0.693 1 47 I 804 i 0.3333333333 : I 0.3333333333 ar 41 152263 : 3 I37I742I ar 508053 : 5 ioi6ir ar 6272 : 7 896 ihr 77 ' 9 9 0.3347153270 ■0.1115717757 0.2231435513 Slog \2 = 2.079441 5412 loge jO = 2.30258509 134 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 3 1. The tabular logarithms of the system of which lO is the base, are derived from the corresponding Napierian logarithms by means of the relation loge;r = loge lo logio^, whence log^o^ = ^i log^-^ — M . loge^- logeio ** The constant -. , denoted above by M. is called the modulus logeio' ^ of common logarithms. Taking the reciprocal of loggio, com puted above, we have M — 0.43429448. The Developments of the Sine and the Cosine, 132. Let f{x) = sin x, then f\x) — cos Xyf'\x) — — sin x,f"'{x) = — cos x,f''^{pc) = sin ;ir ; f'^ being identical with/, it follows that these functions recur in cycles of four ; their values when x — O are o, 1,0, — I, etc. Hence substituting in equation (i). Art. 128, we have x"^ ^ x^ x'' , . sm X = X + • • • . . (i) I-2-3 1-2 • • • 5 I-2- • • 7 ^ ^ In a similar manner, we obtain x^ X* x' cos ;ir = I + + . . . . . (2) 1.2 1-2 3-4 1-2 •• -6 ^ ^ § XIX.] EXAMPLES. 135 Examples XIX. / I. Expand (i + xY- (i + xr= I + mx H ^ -X H ^ ^ -x^ -f . . . 1-2 I-2-3 It is evident that no coefficient will vanish if m is negative or frac- tional. This is the form in which the binomial theorem is employed in computation, x being less than unity. \/ 2. Find three terms of the expansion of sin^ x. sin^ X = x'^ f J 3. Expand tan x to the term involving x'' inclusive. / tan x^^ X -\ 1 h 3 15 / 4. Expand sec x to the term involving x^ inclusive. x^ <.x^ dix^ sec ^ = I + — + — ^ + 7 + 1-2 I-2-3-4 1.2 • • • • 6 5. Expand log sec ^ to the term involving x^ inclusive. X' X X log sec .^ = 1 \ h 2 12 45 / ^ 6. Find four terms of the expansion of £* sec x J o 2X^ f'sec^=i ■\- X ■{■ X -\ h 7. Derive the expansion of log (i — jc') from the logarithmic series, and verify by adding the expansions of log {1 ^ x) and log (i — x)» V 8. Derive the expansion of (i + ^)f* from that of f*. (i + xY' ^\ ^ 2X ^^- • • -h ■— -x'^, ^ ' \'2 1-2 ■■ ' n 136 THE DEVELOPMENT OF FUNCTIONS. [Ex. XIX. 1/ 9 Find, by means of the exponential series, the expansion of xf?'^ including the /zth term. / ^ V 10. Expand — — — by division, making use of the exponential I "T <^ series. ^ x^ X^ T.X^ I \x^ y~^—= I + + V". + • • • I + ^ 2 3 8 ' 30 II. Find the expansion of f'log(i + x) to the term involving^*, by multiplying together a sufficient number of the terms of the series for e' and for log (i + x). 5 ..3 6 6Mog(i + ^) -^ + — + - + ^^- + . . . 2 3 40 v/ 12. Expand log (i + ^'). log(i + f') = log2+- + -3-- — + 13. Expand (i + s')" to the term involving x^ inclusive. (i +£*)«= 2"^! -h^-x + n{n + i) o(^ / j^-v^-ti) N*** , n(7^ + n + 2) x^ t. ^ ^j. i_ -J- . - — ■■} 14. Find the expansion of V{i ± sin 2jc), employing the formula 4^(1 ± sin 2x) = cos X ± sin x. V{i ± sin 2x) = I ±x - -— =F -^ + • • • 1-2 1.2.3 15. Find the expansion of cos" j^t: by means of the formula cos'^.x: = J(i 4- cos 2x). , 2 2'X* 2\x'' cos'^ = I — ^^ H f- . . . I-2-3-4 1-2 ... 6 § XIX.] EXAMPLES. 137 i/ 16. Find the expansion of cos' x^ by means of the formula cos' X = J(cos sx + $ cos x). 1*2 4 I-2-3-4 4 1-2 • • • 2« A''^ 17. Compute loge3, and find log;o3 by multiplying by the value of il/ (Art. 166). 18. Find loge269. J*uf n — 270 = 10 X 3", and /i = — i. 19. Find log, 7, and log. 13. loge3 = 1. 0986 1 23. Iogio3 = 0.477 1 2 13. loge269 =: 5.5947 1 14. I0ge7 = I.9459IOI. l0geI3 = 2.5649494. CHAPTER VIII. Curve Tracing. XX. Equations in the Form y = f(x). 133. When a curve given by its equation is to be traced, it is necessary to determine its general form especially at such points as present any peculiarity, and also the nature of those branches of the curve, if there be any, which are unlimited in extent. The general mode of procedure, when the equation can be put in either of the forms, y =f{x) ox x = #(/), is indicated in the following examples. Asymptotes Parallel to the Coordinate Axes, (34. Example \. -ay — xy=a'' (i) Solving for J/, we obtain ^ = ^:^' • • • (2) WhGnx = o,y = a. Numerically equal positive and nega- tive values of x give the same values for jj/; the curve is there- fore symmetrical with reference to the axis of y. As x increases § XX.] ASYMPTOTES PARALLEL TO THE AXES. 139 from zero, y increases until the denominator, a^ — x^, becomes zero, when y becomes mfinite ; this occurs when x = ± a. Draw the straight Hnes x = ± a. These are hnes to which the curve approaches indefinitely, for we may assign values to X as near as we please to -\- a ov to — a, thus determining points of the curve as near as we please to the straight lines x= a and X = — a. Such lines are called asymptotes to the curve. When X passes the value a,y becomes negative and decreases numerically, ap- proaching the value zero as x increases indefinitely. Hence there is a branch of the curve below the axis of x to which the lines x = a and y =^ o are asymptotes. The general form of the curve is in- dicated in Fig. 15. The point (o, a) evidently corresponds to a minimum ordi- nate. Fig. 15. 135. Example 2. a^x =y {x — of (i) Solving for^, y = {X - a)^ (2) When X is zero, y is zero ; y increases as x increases until X = ay when y becomes infinite. Hence X = a is the equation of an asymptote. When x passes the value <2, y does not change sign, but remains positive, and as x in- creases y diminishes, approaching zero as x increases indefi- nitely. I40 CURVE TRACING. [Art. 1 35. Examining now the values of j which correspond to nega- tive values of x^ we perceive that, y becoming negative, the branch which passes through the origin continues below the axis of x, and that y approaches zero as the negative value of x Fig. 17. increases indefinitely. Hence the general form of the curve is that indicated in Fig. 17. (36. To determine the direction of the curve at any point, we have , dy ^ a 4- X , . The direction in which the curve passes through the origin IS given by the value of tan (/> which corresponds to x = o. From (3), we have ^^Jo dx_ the inclination of the curve at the origin is therefore 45'' Minimum Ordinates and Points of Inflexion. (37. To find the minimum ordinate which evidently exists he left of the ax to zero, and deduce dy on the left of the axis of j, we put the expression for -^ equal x — — a. The minimum ordinate is therefore found at the point whose abscissa is — ^ ; its value, obtained from equation (2), is d'^v K point of inflexion is a point at which ^, changes sign (see § XX.] POINTS OF INFLEXION. I4I Art. 74) ; in other words, it is a point at which tan ^ has a maximum or a minimum value. In this case there is evidently a point of inflexion on the left of the minimum ordinate. From equation (3) we derive dx' " {x-aY putting this expression equal to zero to determine the abscissa, and deducing the corresponding ordinate from (2), we obtain, for the coordinates of the point of inflexion, X = — 2a^ and y — —\a. Oblique Asymptotes, \ZZ, Example I. x' — 2A:y — 2x' —Sy = O, c .. . , (l) Solving this equation for j, we have ^=J'x^-T4 •• — -•. (2) It is obvious from the form of equation (2) that the curve meets the axis of x at the two points (o, o) and (2, o). Since y is positive only when ;r > 2, the curve lies below the axis of X on the left of the origin, and also between the origin and the point (2, o), but on the right of this point the curve lies above the axis of x. 139 Developing the second member of equation (2) into an expression involving a fraction whose numerator is lower in degree than its denominator, we have ,= J^-I+2^,^. . . ... (3) 142 CURVE TRACING. [Art. 139. The fraction in this expression decreases without limit as x in- creases indefinitely ; hence the ordinate of the curve may, by increasing jc, be made to differ as little as we please from that of the straight line This line is, therefore, an asymptote. The fraction 2 — X X' + 4 is positive for all values of X less than 2, negative for all values of x greater than 2, and does not be- come infinite. The curve, therefore, lies above the F^^" ^^' asymptote on the left of the point (2, o), and below it on the right of this point, as represented in Fig. 18. 14-0. There is evidently a minimum ordinate between the origin and the point (2, o). We obtain from equation (2) and df _ dx ~ d^ _ dx" ~ x^ ^ \2x 16 vi''4- (yx^-\- \2x — (^' + 4)' (4) (5) dy _ Putting -j^ = o, we obtain x — O and x =^ 1.19 nearly, the only real roots; the abscissa corresponding to the minimum ordinate is therefore 1.19, the value of the ordinate being about — o.ii. The root zero corresponds to a maximum ordinate at the origin. §XX.] OBLIQUE ASYMPTOTES. 143 Putting (Pi O, we obtain the three roots ;ir = — 2, and 4r = 2 (2 ± V 3) ; the corresponding ordinates are obtained from equation (3). There are, therefore, three points of in- flexion, one situated at the point (— 2, — i), and the others near the points (0.54, — 0.05), and (7.46, 2.55). The inchnation of the curve is determined by means of equation (4) to be tan- 'J at the point (2, o), and tan"' J at the left-hand point of inflexion. (41. Example 4. x" — xy ^ Solving for/ .r' + I ^= — :::— Curvilinear Asymptotes, =- o. , , = x' -\- (0 (2) In this case, on developing jj/ in powers of x, the integral portion of its value is found to contain the second power of x ; the fraction approaches zero when x increases indefinitely ; hence the ordinate of this curve may be made to differ as little as we please from that of the curve j^ = x' (3) The parabola represented by this equation is accordingly said to be a curvilinear asymp- tote. It is indicated by the dotted line in 19. Fig 142. The sign of the fraction - is always the same as that of x, and its value is infinite when X is zero ; hence the curve lies below the parabola on the left of the axis of 7, and above it on the right, this axis being an asymptote, as indicated in Fig. 19. Fig. 19. 144 CURVE TRACING. [Art. 1 42. Taking derivatives, we obtain ^ = ^^~^ (4) and g=.(n.i,). ...... (5) There is a point of inflexion at ( — i, o) ; the inclination of the curve to the axis of x at this point is tan ~' (— 3). There is a minimum ordinate at the point (^^4, \^2). This cubic curve is an example of the species called by- Newton the trident. The characteristic property of a trident is the possession of a parabolic asymptote and a rectilinear asymptote parallel to the axis of the parabola. Examples XXVI. J I. Trace the curve j == x (^^ — i). Since J is an odd function of x^ the curve is symmetrical with re- ference to the origin as a centre. Find the point of inflexion, and the minimum ordinate. 2. Trace the curve y (jr — i ) = ^'. The curve has for an asymptote the line :r = i ; there is a mini- mum ordinate at (2, 2), and a point of inflexion at (4, | i/3). >l 3. Trace the curve y" — x" {x — a), determining its direction at the points at which it meets the axis of x. The asymptote is found by the method of development, thus the equation of the asymptote is therefore § XX.] EXAMPLES. 145 4. Trace the curve x^ ^ xy ^- 2x — y = o. 5. Trace the curve y" = x" -\- x^^ and find its direction at the origin. The curve has a maximum ordinate at (— |, ± f 1/3). The value of y may be taken as the function whose maximum is required. (See Art. 108.) 6. Trace the curve j; = ^' — xy. Find the point of inflexion, the minimum ordinate, and the asymptotes. The curve has a rectihnear asymptote x— — i, and a curvihnear asymptote ji' =^"^ — ^ 4- i. This curve is a trident. (See Art. 142.) 7. Trace the curve y^ = jc^ — x^. Both branches of the curve are tangent to the axis of x at the origin. 8. Trace the curve jv' —/^y = x* + y^. Solving for^, we obtain y^2 ± s/{x^ + ^' + 4) = 2 ± i/[(jc + 2) (x' - ^ + 2)]. The factor x^ — x -^ 2 being always positive, the curve is real on the right of the line x = — 2. Find the points at which the curve cuts the axis, and show that the upper branch has a maximum ordinate for ::c = — f and a mini- mum ordinate for x — o. 9. Trace the curve {x — 2a) xy = a(x — a) (x — 3^). 10. Trace the curve {x — 2a) xy- = a" {x — a) (x — ^a). 11. Trace the curve/ = ^* (i — jv*)' : find all the points at which the tangent is parallel to the axis of x. 12. Trace the curve 6x (i — x) y = i + 3^. This curve has a point of inflexion, determined by a cubic having only one real root, which is between — i and — 2. Find the three asymptotes, and the maximum and the minimum ordinate. 146 CURVE TRACING. [Ex. XX. 13. Trace the curve ^y — {x -^ 2f {1 •\- x^). Solving the equation for j', we have y=± ^4/(i+^')=±(2+.r)(i+i^y. The asymptotes are jj; = a; + 2, j = — .r — 2, and x =0. The curve has a minimum ordinate corresponding to a = .y/2 ; the inclina- tion at the point at which it cuts the axis of x is tan"' (±^1/5). There is a point of inflexion corresponding to the abscissa x = — 6.1 nearly. XXL Curves Given by Polar Equations, 143. The following examples will illustrate some of the methods employed when the curve is given by means of its polar equation. Example t^, r = acosd cos 26. . . . . , . . . (i) When 6 = o^r — a^ the generating point P therefore starts from A on the initial line. As B increases, r decreases and becomes zero when ^ = 45°, /* describing the half-loop in the first quadrant, and arriving at the pole in a direction having an inclination of 45° to the initial line. When 8 passes 45°, r becomes negative, and returns to zero again when 6 — 90°, P describing the loop in the third quad- rant. As 6 passes 90°, r again becomes positive, but returns to zero when 0= 135°, P describing the loop in the second quadrant. As Ovaries from 135'' to 180°, r again becomes negative, /^de- ^^* scribing the half-loop in the fourth quad- rant, and returning to ^, g XXL] CURVES GIVEN BY POLAR EQUATIONS. I47 In this example if we suppose d to vary from 180° to 360°, P will again describe the same curve, and, since d enters the equation of the curve, by means of trigonometrical functions only, it is unnecessary to consider values of 6 greater than 360°. 144, Putting equation (i) in the form r — a{2 cos^O — cos 6), we derive ^ = a{-6 cos^^ sin 6 + sin 6). du To determine the maxima values of r, we place this derivative equal to zero, thus obtaining the roots sin ^ = o and cos Q = ±\ V6', the former gives the point A on the initial line, and the latter gives the values of which determine the position of the maxi- ma in the small loops. The corresponding values of r are =F - V6. To determine the position of the maximum ordinate, we have from (i) y = rsln = ia sin 4^. The maxima values occur when sin 4^9 =1, and the minima when sin 4^ = — I ; that is, we have maxima when — Itt and when 6 = ^7ty and minima when 6 = |7r and -}7t. 145. In the preceding example the substitution of ^ + tt for 6 changes the sign but not the numerical value of r. When this is the case, ^and 6 +7r evidently give the same point of the curve, and the complete curve is described while varies from o to 7t. If however this substitution changes neither the numerical value nor the sign of r, ^and -h 7t will give points symmetrically situated with reference to the pole ; that is, the curve will be symmetrical in opposite quadrants. 148 CURVE TRACING. [Art. 1 45. Again if the substitution of — ^ for Q does not change the value of r, 6 and — B give points symmetrically situated with reference to the initial line, hence in this case the curve is sym- metrical to this line ; but, if the substitution of — ^ for 6 changes the sign of r without changing its numerical value, the curve is symmetrical with reference to a perpendicular to the initial line. The Determination of Asymptotes by Means of Polar Equations, (46. When r becomes infinite for a particular value of d the curve has an infinite branch, and, if there be a corresponding asymptote, it may be determined by means of the expression derived below. Let By denote a value of d for which r is infinite, and let OB be drawn through the pole, making this angle with the initial line ; then, from the triangle OBP, Fig. 20, we have PB^ rsm{d,-d). Fig. 20. Now, if the curve has an asymptote parallel to OB, it is plain that as 6 approaches 6^ the limiting value of PB will be equal to OR, the perpendicular from the pole upon the asymptote. Hence, if the curve has an asymptote in the direction 6^, the expression OR=\rsm{d, - e)\, which takes the form o^ • o, will have a finite value, and this value will determine the distance of the asymptote from the pole. Fig. 20 shows that when the above expression is posi- tive OR is to be laid off in the direction 6^ — 90°. If upon evaluation the expression for OR is found to be in- §XXI.] ASYMPTOTES. 149 finite we infer that the infinite branch of the curve is para- bolic. 147. Example 6. r — aO" (I) Since r becomes infinite when 6 = i, we proceed to apply the method established in the preceding article for determining the existence of an asymptote. In this case we have [r sin {e^ - 6)] _ r aO' sin(i - <9) -j _ The angle = i corresponds to 57° 18', nearly, and since the expression for the perpendicular on the asymptote is neg- ative its direction is ^j + 90° = 147° 18'; consequently, the asymptote is laid off as in Fig. 21. Numerically equal positive and negative values of 6 give the same values for r ; hence the curve is symmetrical with refer- ence to the initial line. While varies from o to i, r is negative and varies from o to 00, giving the infinite branch in the third quadrant. As d passes the value unity, and increases indefinitely, r becomes positive and decreases, approaching indefinitely to the limiting value a, which we obtain from (i) by making 6 in- finite. Hence the curve describes an infinite number of whorls approaching indefinitely to the circle r — a^ which is therefore called an asymptotic circle. The points of inflexion in this curve are determined in Art. 175. Fig. ±i. I50 CURVE TRACING. [Ex. XXI. i Examples XXI. I. Trace the curve r^=^ a cos" \ 0. Show that, to describe the curve, must vary from o to 3 tt ; also that the curve is symmetrical to the initial line. Find the values of which correspond to the maxima and minima ordinate* and abscissas, the initial line being taken as the axis of x. \l 2. Trace the curve r = a {2 s'lnO — ^ sin'o). Show that the entire curve is described while varies from o to tt^ and that the curve is symmetrical with reference to a perpendicular to^the initial line. 7 3. Trace the curve r = 2 + sm 30. A maximum value of r (equal to 3) occurs at = 30° ; a mini- mum (equal to i) at = 90°. The curve is symmetrical with refer- ence to lines inclined at the angles 30°, 90°, and 150° to the initial line. nI J 4. Trace the curve r = i 4- sin 50. The curve consists of five equal loops. 5. Trace the curve r" = a^ sin 30. The curve consists of three equal loops. / 6. Trace the curve r cos = a cos 2O, The curve has an asymptote perpendicular to the initial line at the distance a on the left of the pole. V 7. Trace the curve r = 2 + sin |0. A m.aximum value of r occurs at = 60°, and a minimum at = 180°. The curve has three double points, one being on the initial line. n 8. Trace the curve r cos 20 = a. The curve is symmetrical with reference to the initial line and with reference to a perpendicular to the initial line. There are four asymptotes. § X X I .] EXAMPLES. 1 5 1 9. Trace the curve r sin 40 = « sin t^^. The curve is symmetrical to the initial line, and has three asymp- totes ; the minimum value of r is \a. 10. Trace the curve r" = c^ cos 26. The curve is symmetrical with respect to the pole since r =. -^a *^ (cos 20) : r is imaginary for values of between \Tt and J^. 3. 3. 11. Trace the curve r* = a^ cos |9. The curve consists of three equal loops, r being real for all values of e. 12. Trace the curve r^ cos Q = ^'^ sin 2i^. The curve consists of two loops and an infinite branch which has an asymptote perpendicular to the initial line and passing through the pole. 29 i^. Trace the curve r = ^ . ^ 20 — 1 Find the rectilinear and the circular asymptote, and also the point of inflexion. XXII. The Parabola of thejwth Degree. (48. The term parabola is frequently applied to any curve in which one of the coordinates is proportional to the n\}ix power of the other, n being greater than unity. The parabola proper is thus distinguished as the parabola of the second degree. The general equation of the parabola of the ;^th degree is usually written in the homogeneous form, {a being positive) ^n-r y — x"" , 152 CURVE TRACING. [Art. 148. The curve passes through the origin and through the point {a^ a), for all values of n. Since 7i >i, the curve is tangent to the axis of x at the origin. (49.' The following three diagrams represent forms which the curve takes for different values of n. When n denotes a fraction, it is supposed to be reduced to its lowest terms. Fig. 22 represents the general shape of the curve when n is an even integer, or a fraction having an even numerator and an odd denominator. Fig. 23 represents the form of the curve when 11 is an odd integer or a fraction with an odd numerator and an odd denominator, Fig. 22. the origin being a point of inflexion. Fig. 24 represents the form of the curve when /^ is a fraction having an odd numerator and an even denominator. In this case y is regarded as a two- valued function, and is imaginary when Fig. 23. Fig. 22 is constructed for the parabola in which ;2 = 4. Fig. 23 is the cubical parabola in which 2 Fig. 24 is the semi-cubical parabola in which n = ^ ; the equation being Fig. 24, or ^2jJ/ ay , 1 The curves corresponding to the general equation y=A-^Bx+Cx'^ Dx'+ . . , Lx" are sometimes cdiWed parabolic curves of the nih. degree. § XXIL] THE CISSOID OF DIOCLES. 153 The Cissoid of Diodes, ISO. Let ^ be a point on the circumference of a circle, and BC a tangent at the opposite extremity of the diameter AB\ let AC be any straight line through A^ and take CP = AD ; then the locus of P is the cissoid. To find the polar equation, AB being the initial line, let DB be drawn, and denote the radius of the circle by a\ then AC —2a ^ecO \ and since ADB is a right angle, AD — 2a cos 8. The polar equation of the locus of P^ A being the pole, is, therefore, r = 2^ (sec ^— cos &) cos 6 or 2a sm' cos (I) 151. To obtain the rectangular equation, we employ the equations of transformation sin 6 = -.y cos c/ = — , r r'' = x'-Vf\ whence, eliminating 6 we obtain y r — 2a ^^, rx and thence the rectangular equation of the curve xix" ^ f) = 2af, . . . or / = 2a — X »!♦?♦•• (2) (3) 154 CURVE TRACING. [Art. 152. The Cardioid. 152. The curve defined by the polar equation r = 2^(1 --cos d) . . (i) is called the cardioid. In Fig. 26, A denotes the pole. The polar equation can also be written in the form Fig. 26. r = 4<2; sin^ \Q (2) Transforming equation (i) to rectangular coordinates, we have for the rectangular equation of the cardioid, {x" + /)2 + A^x {p^ + /) - 4^y = o (3) A point at which two branches of a curve have a common tangent is called a cusp. This curve has a cusp at the origin. The Lemniscata of Bernoulli, 153. The curve defined by the polar equation ^2 = ^2 cos 2^ . . . (i) is called the lemniscata. In Fig. 27 O denotes the pole : a is the semi- axis of the curve. From (i), we have Fig. 27. r^ = «2(cos2^-sin2 6^), § XXII.] THE LEMNISCATA. 155 or r'^a^ — ^ ; whence we have {x" ^ff^a\f-x')^o, (2) the rectangular equation of the lemniscata, referred to its cen- tre and axis of symmetry. If we turn the initial line back through 45°, (i) becomes 7-2 = «2 gjj^ 2^, (3) and the corresponding rectangular equation is (;r2+// = 2^;rj/ (4) When the equation has this form, the coordinate axes are the tangents at the origin. The Logarithmic or Equiangular Spiral, (54. This spiral is defined by the polar equation 1 ... (I) or log r — log a + nO^ the logarithm of the radius vector being ^lo, 28 a linear function of the vectorial angle. It is proved in Art. 168 that this curve cuts its radius vector at a constant angle whose cotangent is n ; hence it is sometimes called the equiangular spiral. The Loxodromic Curve. (55. The track of a ship whose course is uniform is a curve that cuts the meridians of the sphere at a constant angle, and is called a loxodromic curve. 156 CURVE TRACING. [Art. 1 5 5. If we project this curve stereographically upon the plane of the equator the meridians will project into straight lines, and, since in this projection angles are unchanged in magnitude, the projection of the curve will make a constant angle with the projections of the meridians and will therefore be an equiangu- lar spiral. Let d denote the longitude of the generating point mea- sured from the point at which the curve cuts the equator, and C the course ; that is, the constant acute angle at which the curve cuts the meridians, the generating point being supposed to approach the pole as 6 increases. Taking as the pole the projection of the pole of the sphere, the polar equation of the projected curve will be of the form r = ^^«^ (i) in which a is the radius of the sphere, since ^ = o gives r ^= a\ we also have n = — cot C, (2) since the angle whose cotangent is ;/ is the supplement of C (see the preceding article). Denoting by ^ the co-latitude of the projected point we have, by the mode of projection, - = tan i^ ; (3) a and, denoting the corresponding latitude by /, Equation (i) is therefore equivalent to tana7r-^/) = f-^-*^; whence, solving for 6^ we have XXII.] THE LOXODROMIC CURVE. 157 (9 = - tan (T loge tan (Itt - \l) = tan C loge tan (jTr + |/), or, employing common logarithms and expressing (9 in degrees, e° = 131.9284 tan C logxo tan (45° + 1/) • . . (4) T/^e Cycloid, 156. The path de- scribed by a point in the circumference of a circle which rolls upon a straight line is called a cycloid. The curve consists of an unlimited number of branches cor- responding to successive revolutions of the generating circle ; a single branch is, however, usually termed a cycloid. Let O^ the point where the curve meets the straight line, be taken as the origin, let P be the generating point of the curve, and denote the angle PCR by ^. Since PR is equal to the line OR over which it has rolled, 0R = PR = aip, and, from Fig. 29, we readily derive X = a(ip — sin f) 1 y = a {i — cos Jp) J (0 157. These two equations express the values of x and f in terms of the auxiliary variable «/?, and constitute the equations of the cycloid. If desirable, ?/? is easily eliminated from equa- tions (i) and an equation between ;i; and / obtained. Thus, from the second equation, we have 158 CURVE TRACING, [Art. 1 57. COS ^ = ^^y ^ whence sin th = n^^7~j) a a and hence from the first of equations (i) ;r=^cos-' ^ — ^{2ay — y"), , . . . (2) or x= a vers-' - — 1/(2^7 — y)- Equations (i) will in general be found more convenient than equation (2). Thus we easily derive from (i) dy ^ s'mtp dip _ sin tfj dx (i — cos tp) dtp ~ I — costp* whence dy _ d fdy \ _ cos ^p — \ dip _ _ i dx^ ~~ dx \dx J (i — cos ipy dx~ a{\ — cos r^^y ' . 158. The cycloid is frequently referred to the middle point O' or vertex of the curve as an origin, the directions of the axes being turned through 90°. Denoting the coordinates referred to the axes O'X' and O'Vy in Fig. 29, by x' and/', we have y = X — art = a {ip — TT — sin tp), x' = 2a — y — a{i + cos ^-?), or, denoting ^ — tt by ip\ y = a (tp'' + sin tp') x'= a{i — cos tp') (3) In these equations tp' = o gives the coordinates of the ver- tex and tp' = ± Tt gives those of the cusps. § XXII.] THE EPICYCLOID. 159 The Epicycloid, 159. When a circle, tan- gent to a fixed circle exter- nally, rolls upon it, the path described by a point in the circumference of the rolling circle is called an epicycloid. Taking the origin at the centre of the fixed circle, and the axis of x passing through A, (one of the posi- tions of P when in contact with the fixed circle,) a, by rp, and Xy being defined by the diagram, we have, evidently, Fig. 30. a^p = bx .'. X = -^ ^. The inclination of CP to the axis of x is equal to ^ + j, or to — 7 — ip ; the coordinates of /'are found by suotracting the pro- jections of CP on the axes from the corresponding projections of OC', hence X = {a + b) cos ip - b cos — - — ?/? y = {a + b) smip — b sin — - — ip (0 These are the equations of an epicycloid referred to an axis passing through one of the cusps. i6o CURVE TRACING. [Art. 159. Were the generating point taken at the opposite extremity of a diameter passing through P in the figure, the projection of 6P would be added to that of 0C\ the axis of x would in this case pass through one of the vertices of the curve, and the second terms in the above values of x and y would have the positive sign. The Hypocycloid. Fig. 31. 160. When the rolling circle has internal contact with the fixed circle, the curve generated by a point on the circumference is called the hypocycloid^ whether the radius of the rolling circle be greater or less than that of the fixed circle. Adopting the notation used in deducing the equa- tion of the epicycloid we have (see Fig. 31), OC=a-b, and ^ I X = ji^^ The inclination of CP to the negative direction of the axis of X is / a — b. hence the equations of the hypocycloid are X = {a — b) cos f + b cos — ^ — ip y — {a — b) simp — b sin — ^ — ^ (I) § XXIL] THE FOUR-CUSPED HYPOCYCLOID. l6l The Four-Cusped Hypocycloid. Fig. 32. 161. In the case of the hypocycloid when b = \a, the circumference of the rolling circle is one-fourth the circum- ference of the fixed circle, and the curve will have a cusp at each of the four points where the coordinate axes cut the fixed circle, as represented in Fig. 32. On substituting \a for b equations (2) Art. 160 become x—\a cos + \a cos 3^ J = f^ sin ^ — \a sin 3?/? (0 Substituting the values of co^yp and sin 3^ from the for- mulas, cos yp — ^ cos' //' — 3 cos //', and sin 3^ = 3 sin ?/' — 4sin'*^;, we have X = a y a cos' ^) _ a sin' ^) ' (2) whence ;r^ — <3:^ cos^ ^, and y^^ — c^ ^\x^^. Adding, we have x^-\-y^—a^, (3) the rectangular equation of the curve. y CHAPTER IX. Applications of the Differential Calculus to Plane Curves. XXIII. The Equation of the Tangent, (62. The equation of the curve being given in the form y z=zf(x)^ the inclination of the tangent at any point is deter- mined by the equation Hence, if (;ir,, y^ be a point of the curve, the equation of a tangent at (;i:„ y^ will be found by giving to the direction- ratio m^ in the general equation y —yx = m{x — X,), dy the value . ; thus or y-y^=f\x^){x-X^) (2) For example, in the case of the semi-cubical parabola f^ax\ § XXIII.] THE EQUATION OF THE TANGENT. 163 dy „ 3 /^ we have -t~ — \\/ - • ax ^ X The point {a, a) is a point of this curve ; the equation of the tangent at this point is, therefore, •^y — 2x = a. The Equation of the NormaL 163. A perpendicular to the tangent at its point of contact is called a normal to the curve. The coordinate axes being rectangular, the direction-ratio of the normal is the negative reciprocal of that of the tangent ; for the inclination of the normal is \n + ^, and tan(J;r + ^) = _ cot ^. The equation of the normal may, therefore, be written thus — As an illustration, let us take the equation of the ellipse x' f - dy h^x whence —-— r- . dx ay The equation of the normal at any point {x^, y^ of the ellipse is, therefore. :64 APPLICATIONS TO PLANE CURVES. [Art. 1 64. Subtangents and Subnormals, (64. Denoting by s the length of the arc measured from some ds fixed point, — denotes the velocity of P^ the generating point ttZ of the curve ; let PT^ equal to ds, be measured on the tangent at P^ then PQ and ^ 7" will represent dx and dy^ and the angle TPQ will be (<> ; hence t dx ^^ cos ^ = ds sm and ^j: = 4/(^^t'' + df). (2) Fig. 33. 165. The distance PT (Fig. 34) on the tangent line inter- cepted between the point .of contact and the axis of x is sometimes called the tangent, and in like manner the in- tercept PN is called the normal. From the triangles PTR and NPR, we have Fig. 34. /^r^j/cosec^=^J-=;.|/[i +(^)], PN — y sec ^ — y -j- — y y dx I + The projections of these lines on the axis of x, that is TR and RN, are called the sub tangent and the siibnormaL From the same triangles, we have § XXI 1 1.] SUBTANGENTS AND SUBNORMALS. 1 65 dx the subtangent, TR = y cot , Fig. 35. equation (i) becomes _ xdy — ydx _ xdy — ydx - . ^ ~ ds ~ V{dx'-{-dy^) '^^ For example, let us determine / in the case of the four- cusped hypocycloid, X— a cos* //?, yz=z a sin' 7/7. Differentiating, dx = — la cos- ^ sin ?/? dip, and dy — ^a sin'' ^ costpdrp ; 'ivhence ds = ^a sin ip cos ^' ^y^?. Substituting in equation (2) we obtain p = a cos' Jp simp -{- a sin' ip cos ?/? = ^ sin tp cos ?/? = ^(^;rj). To ascertain the direction of / it is necessary to determine l66 APPLICATIONS TO PLANE CURVES. [Art. 1 66. ^. The ambiguity in the value of (f> as determined from the equation tan ^ = -^ may be removed by means of one of the formula^ of Art. 164. Thus, in the present case, we have tan ^ = — tan ^, whence ^ = — ^, or ^=: n ^ ip\ but, since cos ^ = — - = — cos ^, as we must take = (^ + 2^) sm ^ . XXIV. Polar Coordinates. 167. When the equation of a curve is given in polar co- ordinates the vectorial angle 6 is usually taken as the inde- pendent variable ; hence, denoting by s an arc of the curve, it is usual to assume that ds and dd have the same sign ; that is, , ds . . . that -j^ is positive. du In Fig. 36 let PT, a portion of the tangent line, represent ds ; then, producing /-, let the rectangle PT be completed, and 1 68 APPLICATIONS TO PLANE CURVES. [Art. 1 67. let ^ denote the angle TPS\ that Is, the angle between the positive directions of r and s. The re- solved velocities of P along and perpen- dr s dicular to the radius vector are — and -TT-, the latter being the velocity which P would have if r were constant ; that is, if P moved in a circle described with ?- as a radius. Hence we have PS = dr and PR = rdd. From the triangle PST, we derive . , rdS . , rdO ^ dr , , tan^. = ^, smy. = --, cosy. = -^^, . . (i) Fig. 36. J 172. The expression deduced below for the function + im is frequently useful. dtf Differentiating (2), we have , du d^'u 2dp § XXIV.] THE PERPENDICULAR UPON THE TANGENT. I /I hence («4-^)^«=^|, , dr or since du— — —^ y d'^'u _T^ dp ^ ^ 'd&'~f"dr' The Perpendicular upon an Asymptote, 173. When the point of contact P passes to infinity the tangent at P becomes an asymptote, and the subtangent OT coincides with the perpendicular upon the asymptote. Hence {Q^ denoting a value of Q for which r is infinite) the length of this perpendicular is given by the expression — -z- , and like the polar subtangent is, when positive, to be laid off in the direction 0, — 90°. This expression for the perpendicular upon the asymptote is also easily derived by evaluating that given in Art. 146. Thus — Points of Inflexion, 174. When, as in Fig. 37, the curve lies between the tan- gent and the pole, it is obvious that r and / will increase and decrease together ; that is, -~ will be positive. When on the dr other hand the curve lies on the other side of the tangent, ~~- is negative. Hence at a point of inflexion -~ must change sign. 1/2 APPLICATIONS TO PLANE CURVES. [Art. 1 74. Now, since/ is always positive, it follows from the equation deduced in Art. 172 that the sign of this expression is the same as that of hence at a point of inflexion this expression must change sign, Mb. As an illustration, let us determine the point of in- flexion of the curve traced in Art. 147 ; viz., ae" e'-i In this case u = -(i — 6 "") ; therefore . ^ J = i (, _,- _6.-) 6' -e^-e ad' Putting this expression equal to zero, the real roots are and it is evident that, as 6 passes through either of these values, ■- . d'^u . the expression u + -^ cha flexion are determined by d^u the expression u + -j^ changes sign. Hence the points of in e= ± ^2 and r=^~. ^ 2 § XXIV.] EXAMPLES. 173 Examples XXIV. 1. Prove that, in the case of the lemniscata r^ = a* cos 20, ^ = 2e + irt, and |^ = 1 2. Find the subtangent of the lituus r" = — , and prove that the perpendicular from the origin upon the tangent is 3. Find the polar subtangent of the spiral r {t^ + e-^) = a. a c-tf e" — e 4. Find the value of/ in the case of the curve r« = a"" sin nO. I J> = a (sin «0) ' «" • 5. In the case of the parabola referred to the focus r = , prove that /' = ar, I + cos ' ^ ^ 6. In the case of the equilateral hyperbola r* cos 20 = a'f prove that/ = — . 7. In the case of the lemniscata r^ = a^ cos 20, prove that/ = —7 • CL 8. In the case of the elHpse r = — — , the pole being at ^ 1 — e cos 174 APPLICATIONS TO PLANE CURVES, [Ex. XXIV. the focus, determine/. 9. In the case of the cardioid r = « (i + cos 6), prove that r^ = 2ap*. 10. Show that the curve r6 sin = « has a point of inflexion at which r = — . 7t XXV. Curvature. 176. If, while a point P moves along a given curve at the ds rate -^, it be regarded as carrying with it the tangent and normal lines, each of these lines will rotate about the moving point P at the angular rate -— - , ^ denoting the inclination of the tangent line to the axis of x. The point P is always moving in a direction perpendicular to the normal with the velocity — ^. Let us consider the at motion of a point A on the normal at a given dis- tance k from P on the concave side of the arc. While this point is carried forward by the motion ds X of P with the velocity -7; in a direction perpen- ^^' ^ ' dicular to the normal, it is at the same time car- ried backward, by the rotation of this line about P, with the § XXV.] CUR VA TURK. I J 5 velocity --3— ; since this is the velocity with which A would move if the point P occupied a fixed position in the plane ; and the direction of this motion is evidently directly opposite to that of P. Hence the actual velocity of A will be ds J d ds whence PC = p~ -yj (i) a

d f L V^j whence ^ = ^„ and p= ^ In this case ds and ^ were assumed to have the same sign, hence ^ must be taken so as to cause j/ to increase. 182. When x and y are expressed in terms of a third vari- able we employ the formula deduced below. Differentiating both ^;ir and ^ being regarded as variable, we have dx dy — dy d^x dx' dxdy-dyd'x , ^ ■*" \dx) ds {dx' -h dff^ (. whence P = d^ = dx dy ^ dyd'^x ^'^ XXV.] EXAMPLES. 179 Examples XXV. 1. Find the radius of curvature of the cycloid :x: = ^ (^ — sin ^'), j; nz ^ (l — cOS ^). ds Prove that # = J (tt — ^), and use p= —; . p= — 2 V{2ay). 2. Find the radius of curvature of \h.t. parabola y^ = /\ax. Va 3. Find the radius of curvature of the catenary and show that its numerical value equals that of the normal at the same point. See Art. 165. 4. Find the radius of curvature of the semi-cubical parabola af = x\ _ (4^? + ^x^x^ "= 6a 5. Find the radius of curvature of the logarithmic curve y = afr. cy l80 APPLICATIONS TO PLANE CURVES. [Ex. XXV. 6. Find the radius of curvature of the cissoid (2a — x)y _ a Vx (Sa — 3 ^)^ 7. Find the radius of curvature of Xk^Q parabola Vx + Vy = 2 ^a. (^+# ^ Va 8. Find the radius of curvature of the cubical parabola ^ ~ 6a'x 9. Find the radius of curvature of the prolale cycloid x = aip — bsinipj y = a — b cos ^\ _ {a^ + b"^ — 2ab cos tjS)^ ' b(a cos ^ — b) XXVI. Envelopes, 183. The curves determined by an equation involving x and y together with constants to which arbitrary values may be assigned are said to constitute a system of curves. The arbitrary constants are called parameters. When but one of g XXVI.] ENVELOPES. I8l the parameters is regarded as variable, denoting it by a^ the general equation of the system of curves may be expressed thus : f{x,y,a)=o (i) When the curves of a system mutually intersect (the intersec- tions not being fixed points), there usually exists a curve which touches each curve of the system obtained by causing the value of a to vary. For example, the ellipses whose axes are fixed in position, and whose semi-axes have a constant sum, constitute such a system ; and, if we regard the ellipse as varying continuously from the position in which one semi-axis is zero to that in which the other is zero, it is evident that the boundary of that por- tion of the plane which is swept over by the perimeter of the varying ellipse is a curve to which the ellipse is tangent in all its positions. A curve having this relation to a given system of curves is called the envelope of the system. Every point on an envelope may be regarded as the limit- ing position of the point of intersection of two members of the given system of curves, when the difference between the cor- responding values of a is indefinitely diminished. For this reason, the envelope is sometimes called the locus of the ultimate intersections of the curves of the given system. 184. If we differentiate equation (i) of the preceding arti- cle (regarding a: as a variable as well as x and y) the resulting equation will be of the general form /i (-^^ J> oc) dx + fj^x, y, a) dy -}- fl{x, y, a) da = o. . (2) In this equation each term may be separately obtained by differentiating the given equation on the supposition that the quantity indicated by the subscript is alone variable. See Art. 64, 1 82 APPLICATIONS TO PLANE CURVES. [Art. 1 84. From equation (2) we derive dy ^ _ /j {x, y, a) f^ {x, f, a) da dx f'y {x, y, a) f'^ {x, y, a) dx * * * * ^3; In Fig. 39 let PC be the curve corresponding to a particular value of or, and let P be the point {x, y) ; then ^ the expression for -j- given in equation (3) determines the direction in which the point P is actually moving when x^ y, and a vary ^ simultaneously. This direction depends there- Fig. 39. fore in part upon the arbitrary value given to the ratio -7- . 185. Now if a were constant da would vanish, and equa- tion (3) would become ^ = _ /^(^>-^>^) (.\ dx f;{x,y,ay •••••• W This expression for -~ determines the direction in which P moves when PC is a fixed curve. Let AB be an arc of the envelope, and let C be its point of contact with PC, Now, if P be placed at the point C, it is obvious that it can move only in the direction of the common tangent at C^ whether a be fixed or variable. It follows there- fore that, at every point at which a curve belonging to the system touches the envelope, the expressions for -— given in equations (3) and (4) must be identical in value. Assuming that f'^ (x, y, a) and /' (x^ y, a) do not become in- finite for any finite values of x andjj/, the above condition re- quires that /;(-^;/;«') = (5) § XXVI.] ENVELOPES. 183 Hence the coordinates of every point of the envelope must satisfy simultaneously equations (5) and (i) ; the equation of the envelope is therefore obtained by eliminating a between these two equations. 186. Let it be required to find the envelope of the circles having for diameters the double ordinates of the parabola If we denote by a the abscissa of the centre of the variable circle, its radius will be the ordinate of the point on the para- bola of which a is the abscissa, the equation of the circle will therefore be y"^ + {x — a)- — ^aa — O (l) Differentiating with reference to the variable parameter «', we have — 2 (,r — n') — 4^ = o, or a = 2a -]r X \ (2) substituting in (i), and reducing, we obtain /^4a{a + x). . (3) The envelope is, therefore, a parabola equal to the given para- bola and having its focus at the vertex of the given parabola. Two Variable Parameters. i87. When the equation of the given curve contains two variable parameters connected by an equation, only one of these parameters can be regarded as arbitrary, since, by means of the equation connecting them, one of the parameters can be eliminated. Instead, however, of eliminating one of the parameters at once, it is often better to proceed as in the fol- lowing example. 1 84 APPLICATIONS TO PLANE CURVES. [Art. 1 8/. Required, the envelope of a straight line of fixed length a, which moves with its extremities on two rectangular axes. Denoting the intercepts on the axes by a and yS, the equa- tion of the line is a and fi being two variable parameters which, by the condi- tions of the problem, are connected by the relation a''-^ §^ = c?. . (2) Differentiating (i) and (2) with respect to a and § as vari- ables, we have xda ydfi o? fi' 2-4-^ = 0, (3) and ada + jSd^ = o . . (4) We have now four equations from which we are to eliminate dcx. a^ y5, and the ratio — . Transposing and dividing (3) by (4), a p we obtain X _ y Substituting in (i) the value of y derived from the I'ast equation, we have whence by equation (2) a = x^a^. § XXVI.] EVOLUTES. 185 In like manner we find Hence, substituting in (2) — - The envelope is therefore a four-cusped hypocycloid. Evolutes, 188. In Fig. 40 let C be the centre of curvature of the given curve : this point is so determined (see Art. 176) as to have no motion in a direction perpendicular to the normal PC^ but since p is in general variable, it has a mo- tion in the direction PC, Hence C describes a curve to which the normal PC is always tangent at the point C. Moreover, since P has no motion in the direction PC^ if we regard P as a fixed p-j^ .^^ point on this line, the rate of C along this moving line will be identical with its rate along the curve which it describes. Hence the motion of PC is the same as that of a tangent line rolling upon the curve described by (7, while P, a fixed point of this tangent, describes the original curve. The curve described by the centre of curvature C is there- fore called the evolute of the curve described by P, and the latter is called an involute of the former. 189. Since the evolute of a given curve is the curve to which all the normals to the given curve are tangent, it is evidently the envelope of these normals. 1 86 APPLICATION'S TO PLANE CURVES. [Art. 1 89. The equation of the normal at the point (^, y) of a given curve may be written in the form x'-x + {y -y)£^=o, (i) (jr',y) being any point of the normal. See Art. J63. In this equation y and -^ are functions of x determined by dX the equation of the given curve, and x is to be regarded as the arbitrary parameter. Hence, differentiating with reference to ;r, we have The equation of the evolute is therefore the relation be- tween x' and y which arises from the elimination of x between equations (i) and (2). 190- As an illustration, let it be required to find the evolute of the common parabola y = 2a^x^ ; dy _ /a\^ dx~\^)' whence -/=(-) , and d^y dx \x/ dx^ 2x\ Substituting, we obtain from equation (2) of the preceding article x^= — ia^y ; whence, from equation (i) of the same article, 2yay^ = 4(x' — 2df, the equation of the evolute, which is, therefore, a semi-cubical parabola having its cusp at the point (2^, o). § XXVI.] EXAMPLES. 187 191. It is frequently desirable to express the equation of the normal in terms of some parameter other than x before differentiating. Thus, let us determine the evolute of the ellipse by means of the equation of the normal in terms of the eccen- tric angle. The equations of the eUipse are X — a cos ^', and jj/ = ^ sin ^ ; whence dx= — a sin tp dtp, and dy — b cos rp dtp. Substitution in the equation of the normal, {x — x)dx + (y — y) dy — o, gives ax' sin ^ — by' cos ^ — (c^ — Ij^) sin ^ cos ^ = o. Differentiating, we have ax' cos tp + by sin tp — {0? — b^) (cos^ tp — sin^ tp) z= o; eliminating y' and x^ successively, and dropping the accents, ax = {d^ — IP) cos^ tp and by = — {c? — ^) sin* tp ; whence {ax'f + {byf = (a^ - IPf, Examples XXVI. I. Find the envelope of the system of parabolas represented by the equation y ■=—{x-a\ in which a is an arbitrary parameter and c a fixed constant. 1 88 APPLICATIONS TO PLANE CURVES. [Ex. XXVI. 2. Find the envelope of the circles described on the double or- dinates of an ellipse as diameters. '2 I 7,2 ^' a" + ^' b' 3. Find the envelope of the ellipses, the product of whose semi- axes is equal to the constant ^^ The conjugate hyperbolas, 2xy = ± <^^ 4. Find the envelope of a perpendicular to the normal to the para- bola, y = 4aXy drawn through the intersection of the normal with the axis. y = 4a {2a — x). 5. Find the envelope of the ellipses whose axes are fixed in posi- tion, and whose semi-axes have a constant sum c. The f our-cusped hypocycloid, x^ + y^ = fi, 6. Given the equation of the catenary prove that / ^ ^\ y'= 27, and x'= x — — Is"^ — € «j, and deduce the equation of the evolute. ^'= a log - >'' ± (y" - 4'^')^ ^ y (y. _ ^')i. 2a 4a 7. Derive the equation of the evolute to the hyperbola, its equa- tions in terms of an auxiliary angle being X = asectp and y = b tan tp. § XXVI.] EXAMPLES. 189 The equation of the normal is ax sin ip ^ by= {(^ -\- b^^ tan ^, and the equation of the evolute is ^!^l — b^y^ — {a" + ^')3. 8. Find the equation of the evolute of the cycloid. The equation of the normal is sin if) , X -{■ y — 7 — atp — o. I — cos ip The equations of the evolute are . :x: = ^ (^ + sin ^) and y = — a{i — cos^). The evolute is therefore a cycloid situated below the axis of x, having its vertex at the origin. See equations (3), Art. 158. CHAPTER X. Functions of Two or More Variables XXVII. The Derivative Regarded as the Limit of a Ratio, 192. The difference between two values of a variable is fre- quently expressed by prefixing the symbol A to the symbol denoting the variable, and the difference between correspond- ing values of any function of the variable, by prefixing z/ to the symbol denoting the function. Hence x and x + Ax denote two values of the independent variable, and Af(x) denotes the difference between the corresponding values oi f{x)\ that is, Ay = /f{_x)=f{x-^ Ax>)-f{x). ... (I) If we put Ax = o, we shall have Ay = o; hence the ratio ^_ ^f{x ^ Ax) - /{x) Ax Ax ^ ' takes the indeterminate form - when Ax = o. The value as- o sumed in this case is called ^/le limiting value of the ratio of the increments, Ay and Ax, when the absolute values of these incre- ments are diminished indefinitely. 193. To determine this limiting value, for a particular value a of X, we put a for x and z for Ax in the second member of § XXVII.] THE DERIVATIVE REGARDED AS A LIMIT. IQI equation (2), and evaluate for ^ = o, by the ordinary process (see Art. 82). Thus Z Jo Therefore when Ax is diminished indefinitely, the limiting value of — corresponding to ;ir = ^ Ax value of Xj we have in general of — corresponding to ;ir = ^ is -j- , and, since <3: denotes any Ax ax_\ a limit of -J- — -j~. Ax ax A V If we denote by e the difference between the values of -~ and ^ Ax —-. we shall have ax %'%*■• w and the result established in the preceding article may be ex- pressed thus — ^ = o when Ax — 6 ; in other words, e is a quantity that vanishes with Ax, Partial Derivatives, 194. Let t^=^fix,y\ in which x and y are two independent variables. The deriva- tive of u with reference to x, y being regarded as constant, is denoted by ~-z- u, and the derivative of u with reference to r, x ax being constant, by ~j- u. These derivatives are called the par" ay tial derivatives of u with reference to x and y respectively. 192 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 1 94. Adopting this notation, the result established in Art. 64 may be expressed thus ; du = — - u ' dx H — — U' dy' dx dy provided u denotes a function that can be expressed by means of the elemejttary functions differentiated in Chapters II and III. It is now to be proved that this result is universally true. 195. Let AxU denote the increment of ti corresponding to Ax^ y being unchanged, AyU the increment corresponding to Ay, X being unchanged, and Au the increment which u receives when x and y receive the simultaneous increments Ax and Ay, Let u ~f{x + Ax,y)y and m" =f{x + Ax, y + Ay) ; then A.xU ^^ u ~ u, AyU = u'' — u\ and Au = u" — u\ hence Au = AxU -k- Ayu' (i) Denoting by At the interval of time in which x, y, and u re- ceive the increments Ax, Ay, and Au, we have Au __ AxU AyU' , . Since Au Is the actual increment of u in the interval At, the du limit of the first member of equation (2) is, by Art. ig^, — , the ai A u rate of u. The limit of -^r- is the rate which u would have At § XXVII.] PARTIAL DERIVATIVES. I93 were x the only variable ; and, since -j-u- dx \s the value which du assumes when this supposition is made, if we put -r-u • dx = dj:U. ~~ ~ dx d u this rate will be denoted by --J— . Hence by equation (4), Art. 193, equation (2) becomes du , dj^u . , dyu' „ in which e, e\ and e" vanish with At ; but when At = o, Ax = o, and therefore u' = u ; hence, putting At = o, we have du dt dxU dyU " dt ^ dt' du =:d^u -\- dyU\ Therefore that is, du — -j-U'dx-\--r-U' dy, dx dy 196. This result is usually written in the form , du J du - du= -r-dx ^- -j-dy, dx dy but when written in this form it must be remembered that the fractions in the second member represent partial derivatives^ the symbol du in the numerators standing for the quantities denoted above \yy d^cU and dyU, which are sometimes Q.d\\^ A par- tial differentials. The du that appears in the first member is called the total differential of u when x and y are both variable. The above result is easily extended to functions of more than two independent variables. \(^^ FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXVII. Examples XXVII. 1. Given u — (x^ + J^*)^, prove that du du _ XV 2. Given u = — =^-— , prove that v X -\- y du du _ dx dy ' 3. Given w = tan"' f — l~'i'> P^ove that du , du x-j- +y-j- = 0. dx dy 4. Given u = logyX^ to find md — . ^ __ I ^ du _^ ^ logjp ^/^e ^ log j; * dy~~ y (log^')^ XXVIII. The Second and Higher Derivatives regarded as Limits. 197. In Art. 193 it is shown that Jy _ dy Ax dx In this equation ^ is a function of x and likewise of Ax-, hence de the derivative -j- is in general a function of x and of Ax. It is § XXVIIL] THE SECOND DERIVATIVE AS A LIMIT. I95 also proved in the same article that e becomes zero when Ax vanishes; that is, e assu7ttes a constant value independent of the value of X when Ax becomes zero ; hence, when Ax is zero, the derivative of e with reference to x must take the value zero, whatever be the value of ;r ; in other words, —r- vanishes with Ax, dx In a similar manner it may be shown that each of the higher derivatives of e with reference to x vanishes when Ax — O. 198. Since ~ is a function of ;r, A -f- will denote the incre- Ax Ax ment of this function corresponding to Ax, Employing the symbol —— to denote the operation of taking this increment, and dividing the result by Ax, we obtain, by applying to this function the principle expressed in equation (4), Art. 193, A Ay d Ay , , . Ax Ax dx Ax ^ ^ -m-'Y^ d'y ... de^ dx^ ' dx de In this equation both e' and --j- vanish with Ax by the preced- ing article ; hence the sum of these quantities likewise vanishes with Ax^ and may be denoted by e. Thus we write ^.^=J> + ,. (2) Ax Ax dx^ ^ ^ 196 FUNCTIONS OF TWO OR MORE VARIABLES, [Art. I99. 199. Since Ax is an arbitrary quantity it may be regarded Ay as constant, whence A -~ is the increment of a fraction whose Ax denominator is constant ; but this is evidently equivalent to the result obtained by dividing the increment of the numerator by the denominator ; that is, J 4r^ A'Ay ^ Ax Ax The numerator A - Ay is usually denoted by the symbol A^y\ hence equation (2) may be written thus : -^ = ^-^ + . . (3) Ax' dx'^'' ^^^ and, since e vanishes with Ax^ it follows that the second deriva- tive is the limit of the expression in the first member of equa- tion (3). In a similar manner it may be shown that each of the higher derivatives is the limit of the expression obtained by substi- tuting A for d in the symbol denoting the derivative. Higher Partial Derivatives, 200. The partial derivatives of u with reference to x and y are themselves functions of x and y. Their partial derivatives, viz., d du d dti d du a ^ ^^^ dx dx"* dy dx'' dx dy^ dy dy^ are called partial derivatives of u of the second order. It will now be shown that the second and third of these derivatives, although results of different operations, are in fact identical ; that is, that d du _ d du dy dx dx dy* § XXVI 1 1.] HIGHER PARTIAL DERIVATIVES. 1 97 Employing the notation introduced in Art. 195, we have A^u=f{x + Ax,y)- f{x,y)', if in this equation we replace y hy y + Ay, we obtain a new value of Aj^u, and, denoting this value by A'ji, we have A'ji =f{x+ Ax, y + Ay) - f {x, y + Ay). Since this change in the value of A^u results from the increment received by y, the expression for the increment received by A^u will be Ay (A^^u) ; hence Ay {Aj,u) = A'^u — Aj,u, or Ay {A^7c)=f{x + Ax, y^Ay)-f{x,y + Ay)-f{x + Ax, })-^f{x,y). The value of A^^Ayu), obtained in a precisely similar manner, is identical with that just given ; hence Ay{A,u) = A^{AyU) (I) Since Ax is constant, we have, as in Art. 199, ^y {^:cU) ^ ^ ^ A^ Ax ^ ' Ax' Hence, dividing both members of equation (i) by Ax • Ay, we have Ay ^ A^u _ 4r AyU , . Ay' Ax Ax' Ay' ^ ' or, employing the symbol — as in Art. 198, A A A A Ay Ax Ax Ay 198 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 200. From this result, by a course of reasoning similar to that em- ployed in Art. 198, we obtain d du d du f . — = — . — - (3) dy dx dx dy 201. The partial derivatives of the second order- are usually denoted by ^u d\i dHc^ dx'* dxdy' dy'' the factors dx and dy in the denominator of the second being, by virtue of formula (3), interchangeable, as in the case of an ordinary product. The numerators of the above fractions are of course not identical. Compare Art. 196. Formula (3) of the preceding article is readily verified in any particular case. Thus, given u=.y^y . du . . du ^ ^ whence -r- = y^^og-y, and — - = xy"^'^ ; dx ^ dy d du , , . d du -.-=^^-(.l0g^+l):=^.^. nxamples XXVIII. 1. Given u = sec {y + ax) + tan {y — ax), prove that ^""^ d/' 2. Verify the theorem , , = , , when u = sin (jc/). ^ dxdy dyax 3. Verify the theorem ^ = -j^ ^^^^ ^ ~ ^^^ ^^^ ^^^ "^ •^'^' § XXVIII.] EXAMPLES. 199 4. Verify the theorem , „ . = — — r^ when u = tan — . ay ax ax ay y 5. Verify the theorem ^ = ^ when «^ = jj/ log (i •{- ^\ dy dx^ dx^ dy 6. Given 2^ = sinjc cosj, prove that d^u d*u d*u df dx^ dx^ dy^ dxdydxdy* 7. Given u = x^z* -\- s'y'^z^ + x^y^z^^ derive d'u dx^ dy dz = ^y^z"" + 2>yz. 8. Given i/(4^^-^) , prove that //' d"" dadb 9. Given « = (^ + jj;)', prove that d^'^?/ ^/V _ du dx^ dx dy~ dx' 10. Given u j7, prove that ^^2^ ^2^ ^^ _ d^^lf^~d?~'^' AN ELEMENTARY TREATISE INTEGRAL CALCULUS FOUNDED ON THE xMETHOD OF RATES OR FLUXIONS WILLIAM WOOLSEY JOHNSON PROFESSOR OF MATHEMATICS AT THE UNITED STATES NAVAL ACADEMY ANNAPOLIS MARYLAND FOURTH THOUSAND. NEW YORK: JOHN WILEY AND SONS, 53 East Tenth Street, 1893. CoPVKtGHT, 1881, By JOHN WILEY AND SONS. PHESS OF J. J. LITTLE L CO., NOS. 10 TO to A3T0R PLACE, NEW YORK. PREFACE. This work, as at present issued, is designed as a shorter course in the Integral Calculus, to accompany the abridged edition of the treatise on the Differential Calculus, by Pro- fessor J. Minot Rice and the writer. It is intended hereafter to publish a volume commensurate with the full edition of the work above mentioned, of which the present shall form a part, but which shall contain a fuller treatment of many of the sub- jects here treated, including Definite Integrals, and the Me- chanical Applications of the Calculus, as well as Elliptic Inte- grals, Differential Equations, and the subjects of Probabilities and Averages. The conception of Rates has been employed as the foundation of the definitions, and of the whole subject of the integration of known functions. The connection be- tween integration, as thus defined, and the process of summa- tion, is established in Section VII. Both of these views of an integral — namely, as a quantity generated at a given rate, and as the limit of a sum — have been freely used in expressing geometrical and physical quantities in the integral fo^-fr. HI IV PREFACE, The treatises of Bertrand, Frenet, Gregory, Todhunter, and Williamson, have been freely consulted. My thanks are due to Professor Rice for very many valuable suggestions in the course of the work, and for performing much the larger share :)f the work of revising the proof-sheets. -W. W. J. U. S. Naval Academy, July, i88i. CONTENTS. CHAPTER 1. Elementary Methods of Integration. I. PAGE Integrals - . . I The differential of a curvilinear area 3 Definite and indefinite integrals 4 Elementary theorems 6 Fundamental integrals. ... 7 Examples I lo II. Direct integration ..,,..,, 14 Rational fractions 15 Denominators of the second degree 16 Denominators of degrees higher than the second 19 Denominators containing equal roots 22 Examples II 26 III. Trigonometric integrals 33 Cases in which sin"^ cos^ Q de is directly integrable 34 The integrals sin* e dd, and cos" ede 36 The integrals f -r-^"— , f ^ , and f J sm cos d J sm J ^, 37 cosei V VI CON-TENTS. PAGE Miscellaneous trigonometric integrals .38 The integration of ; 40 ^ a -\- b cose ^ Examples III ; 43 CHAPTER II. Methods of Integration — Continued. IV. Integration by change of independent variable 50 Transformation of trigonometric forms 51 Limits of a transformed integral 53 The reciprocal of x employed as the new independent variable 53 A power of x employed as the new independent variable 54 Examples IV 56 V. Integrals containing radicals 59 Radicals of the form .^{ax^ + b) 61 dx The integration of — 64 \/{x'^ ± a') Transformation to trigonometric forms 65 Radicals of the form \/(ax'^ + bx + c) t 67 dx , dx ^[{x - a){x ~ fJ)] ^"^ J V[(^-a')(/^-^)] Examples V 70 The integrals I — and f — 68 VI. Integration by parts 77 A geometrical illustration 78 Applications 78 Formulas of reduction 81 Reduction of sin"' e de and ( tion of I si] cos'« ede 82 Reduction of I sin"^ e cos« edQ 84 CONTENTS. Vll PAGE Illustrative examples 87 Extension of the formula employed in integration by parts 8g Taylor's theorem 90 Examples VI 91 VII. Definite integrals * 97 Multiple-valued integrals 100 Formulas of reduction for definite integrals loi Elementary theorems relating to definite integrals 104 Change of independent variable in a definite integral 105 The differentiation of an integral 106 Integration under the integral sign 109 The definite integral regarded as the limiting falue of a sum iii Additional formulas of integration 115 Examples VII 117 CHAPTER III. Geometrical Applications. VIII. Areas generated by variable lines having fixed directions 123 Application to the witch 124 Application to the parabola when referred to oblique coordinates 126 The employment of an auxiliary variable 126 Areas generated by rotating variable lines 12S The area of the lemniscata 129 The area of the cissoid 130 A transformation of the polar formulas 130 Application to the folium 131 Examples VIII 134 IX. The volumes of solids of revolution 141 The volume of an ellipsoid 143 Solids of revolution regarded as generated by cylindrical surfaces 144 Double integration 145 Determination of the volume of a solid by double integration 149 Vlll CONTENTS. PAGE The determination of volumes by triple integration 150 Elements of area and volume 152 Polar elements 154 The determination of volumes by polar formulas 155 Polar coordinates in space 157 Application to the volume generated by the revolution of a cardioid 159 Exaj7iples IX 160 X. Rectification of plane curves 168 Rectification of the semi-cubical parabola 168 Rectification of the four-cusped hypocycloid 169 Change of sign oi ds 1 70 Polar coordinates ! 1 70 Rectification of curves of double curvature 171 Rectification of the loxodromic curve ; 172 Examples X 173 XI. Surfaces of solids of revolution 178 Quadrature of surfaces in general 179 The expression in partial derivatives for sec v i&o The determination of surfaces by polar formulas 181 Examples XI 183 XII. Areas generated by straight lines moving in planes 186 Applications 187 Sign of the generated area 189 Areas generated by lines whose extremities describe closed circuits 190 Amsler's Planimeter 191 Examples XII 193 XIII. Approximate expressions for areas and volumes 195 Simpson's rules 197 Cotes' method of approximation 19S CONTENTS. IX PAGE Weddle's rule 199 The five-eight rule 199 The comparative accuracy of Simpson's first and second rules 2CX) The application of these rules to solids .' ^ 200 WooUey's rule 201 Examples XIII 202 CHAPTER IV. Mechanical Applications. XIV. Definitions 204 Statical moment 204 Centres of gravity 206 Polar formulas 208 Centre of gravity of the lemniscata 209 Solids of revolution 2og Centre of gravity of a spherical cap 210 The properties of Pappus 210 Examples XIV 212 XV. Moments of inertia 219 Moment of inertia of a straight line 220 Radii of gyration 220 Radius of gyration of a sphere 221 Radii of gyration about parallel axes 222 Application to the cone 223 Pola ; moments of inertia 225 Examples XV 225 THE INTEGRAL CALCULUS CHAPTER I. Elementary Methods of Integration. I. Integrals, I. In an important class of problems, the required quanti- ties are magnitudes generated in given intervals of time with rates which are either given in terms of the time /, or are readily expressed in terms of the assumed rate of some other independent variable. For example, the velocity of a freely falling body is known to be expressed by the equation v=gt, ......... (i) in which t is the number of seconds which have elapsed since the instant of rest, and ^ is a constant which has been deter- mined experimentally. If s denotes the distance of the body 2 ELEMENTARY METHODS OF INTEGRATION. [Art. I. at the time /, from a fixed origin taken on the line of motion, V is the rate of s ; that is, ds '' = di' hence equation (i) is equivalent to ds = gtdt, . - . ' (2) which expresses the differential of s in terms of / and dt. Now it is obvious that ^g^^ is a function^of / having a differential equal to the value of ds in equation (2) ; and, moreover, since two functions which have the same differential (and hence the same rate) can differ only by a constant, the most general expression for s is s = iP'+C, . (3) in which C denotes an undetermined constant. 2. A variable thus determined from its rate or differential is called an integral, and is denoted by prefixing to the given differential expression the symbol , which is called the integral sign."'^ Thus, from equation (2) we have -\^ dt. which therefore expresses that i- is a variable whose differential IS gtdt; and we have shown that gtdt = ^gt^ + C. The constant C is called the constant of integration ; its occurrence in equation (3) is explained by the fact that we have not determined the origin from which s is to be measured. * The origin of this symbol, which is a modification of the long j, will be explained hereafter. See Art. 100. § I.] THE DIFFERENTIAL OF A CURVILINEAR AREA. 3 If we take this origin at the point occupied by the body when at rest, we shall have ^ = o when / = o, and therefore from equation (3) 6^=0; whence the eqtiation becomes s — \gt^. The Differential of a CurviUnear Area, 3. The area included between a curve, whose equation is given, the axis of x and two ordinates affords an instance of the second case mentioned in the first paragraph of Art. I ; namely, that in which the rate of the generated quantity, al- though not given in terms of /, can be readily expressed by means of the assumed rate of some other independent variable. Let BPD in Fig. i be the curve whose equation is supposed to be given in the form Supposing the variable ordinate PR to move from the position AB to the position CD^ the required area ABDCis the final value of the Fig. i. variable area ABPR, denoted by ^, which is generated by the motion of the ordinate. The rate at which the area A is generated can be expressed in terms of the rate of the independent variable x. dA dx assumed rates are denoted, respectively, by -—- and -— dt dt The required and the and, to express the former in terms of the latter, it is necessary to express dA in terms of dx. Since x is an independent variable, we may assume dx to be constant ; the rate at which A is gen- erated is then a variable rate, because PR or y is of variable length, while moving at a constant rate along the axis of x. Now dA is the increment which A would receive in the time 4 ELEMENTARY METHODS OF INTEGRATION. [Art. 3. dt, were the rate of A to become constant (see Diff. Calc, Art. 17). If, now, at the instant when the ordinate passes the position PR in the figure, its length should become constant, the rate of the area would become constant, and the increment which would then be received in the time dt, namely, the rectangle PQSRy represents dA. Since the base RS of this rectangle is dx, we have dA—ydx—f{x)dx (1) Hence, by the definition given in Art. 2, A is an integral, and is denoted by A^\^f{x)dx . (2) Definite Integrals. 4. Equation (2) expresses that y^ is a function of x^ whose differential \^f{x)dx ; this function, like that considered in Art. 2, involves an undetermined constant. In fact, the expres- sion f{x)dx is manifestly insufficient to represent precisely the area ABPR, because OA, the initial value of x, is not indi- cated. The indefinite character of this expression is removed by writing this value as a subscript to the integral sign ; thus, denoting the initial value by a^ we write \^f{x)dx, (3) in which the subscript is that value of x for which the integral has the value zero. If we denote th.Q final value of x (OC in the figure) by d, the area. A BDC, which is a particular value of Aj is denoted by § I.] ' DEFINITE INTEGRALS. 5 writing this value of x at the top of the integral sign, thus, ABDC \A^)dx (4) This last expression is called a definite integral^ and ^ and b are called its limits. In contradistinction, the expression f(x)dx is called an indefinite integral, 5. As an application of the general expressions given in the last two articles, let the given curve be the parabola Equation (2) becomes in this case A ={:t^dx. Now, since ^x^ is a function whose differential is x^dx^ this equation gives A={x'dx = j^x^-h C, (I) in which C is undetermined. Now let us suppose the limiting ordinates of the required area to be those corresponding to ;r = i and x = ;^. The vari- able area of which we require a special value is now represented by [ x^dxy which denotes that value of the indefinite integral which vanishes when x = 1. If we put ;ir = i in the general expression in equation (i), namely ^x^ + C, we have ^ + C;- hence if we subtract this quantity from the general expression, we shall have an expression which becomes zero when x = i. We thus obtain A={x^dx=ix'-i. 6 ELEMENTARY METHODS OF INTEGRATION. [Art. 5, Finally, putting, in this expression for the variable area, x = 3, we have for the required area 6. It is evident that the definite integral obtained by this process is simply the difference between the values of the indefinite integral at the upper and lower limits. This difference may be expressed by attaching the limits to the symbol ] affixed to the value of the indefinite integral. Thus the process given in the preceding article is written thus. jydx = -i:^ + cj=9-i = The essential part of this process is the determination of the indefinite integral or function whose differential is equal to the given expression. This is called the integration of the given differential expression. Elementary Theorems, 7. A constant factor may be transferred from one side of the integral sign to the other. In other words, \{ m is a constant and u a function of x^ mudx = m udx. Since each member of this equation involves an arbitrary constant, the equation only implies that the two members have the same differential. The differential of an integral is by definition the quantity under the integral sign. Now the second member is the product of a constant by a variable factor ; hence its differential \'=>md\ \udxV that is, m u dx, which is also the differential of the first member. § T.] ELEMENTARY THEOREMS, 8. This theorem is useful not only in removing constant factors from under the integral sign, but also in introducing such factors when desired. Thus, given the integral recollecting that d{x''-^') - (it + \)x''dx, we introduce the constant factor n + i under the integral sign ; thus, \x''dx — — — \{n + \)x"dx = — — x"-^ ' 4- C 9. If a differential expression be separated into parts ^ its in- tegral is the sum of the integrals of the several parts. That is, if u^ 7', w, ' • ' are functions of x^ \{ii-\-v-\-w-\-''' ')dx = \ic dx -\- \v dx + \w dx + • • • For, since the differential of a sum is the sum of the differ- entials of the several parts, the differential of the second mem- ber is identical with that of the first member, and each member involves an arbitrary constant Thus, for example, (2 — Vx) dx = 2dx — \xdx = 2x — ^x^-\- Cy the last term being integrated by means of the formula deduced in Art. 8. jFundamenla/ Integrals, 10. The integrals whose values are given below are called the fimdamental integrals. The constants of integration are generally omitted for convenience. ELEMENTARY METHODS OF INTEGRATION, [Art. lO. Formula (a) is given in two forms, the first of which is de- rived in Art. 8, while the second is simply the result of putting n——m. It is to be noticed that this formula gives an indeter- minate result when n— — \\ but in this case, formula {U) may be employed.* The remaining formulas are derived directly from the for- mulas for differentiation; except that (y'), (^'), (/'), and {in') X are derived from (y), {k), (/), and {nt) by substituting - for x. n-V\ ^ Jx'" {m — i) x'"-^ ^^ ^ ^ = log(±^)t.-^.(^ . . w i^< H-^^.< w ax X a^'dx cos6 de = sin 6* -<- L- . . Sin^^^rrz _ COS ^ .4* .(L. . (d) * Applying fonnula {a) to the definite integral x^dx^ we have ]a )b A« + 'f ^« + I x'^dx^^- Hf , a n+ I which takes the form - when « — — i ; but, evaluating in the usual manner, = ^— ^^- =log3-loga; n + 1 J« = — I I J« = — I a result identical with that obtained by employing formula (d). f That sign is to be employed which makes the logarithm real. See Diff. Calc, Art. 43. § I.] FUNDAMENTAL INTEGRALS. 9 , I^^H^^^^'""^--^-^ (^^ — ?-^-^ = cosec 6* cot<9^6' = — cosec <9.-V: C«^. . (/) | y(j ^^) = sin-i ;r + (7 = - cos-i x -r C . '. . (y) \ y, 2^ j i\ = sin-^ - + 6^= - cos"^- + C . . . (/') J -/(^^ — ;r) a a ^-^ ' {j^^^ = tRn-'x+ C=---cot-'x+ C, ..... (/^) [-2^ = -tan-i-+ C= - -cot-i-+ (:". . . . (k') }ar + ^ a a a a ^ ' fdx x^i^ - I) ^sec"^^ + (7= - cosec-^;ir + C. . . (/) lO ELEMENTARY METHODS OF INTEGRATION, [Ex. I. Examples I. Find the values of the following integrals [dx i [dx , { dx ' h x^ r dx / 5. Vxdx, ^ 6. [ (x- ifdx, a yj 7. I (a — bxYdXy \ 8. J {a + x^dx, f^'dx [~^dx 2Vx. Vx' X^ + X — ^. 3 a'^x — abx 4- 3 Jo ~ Zb' 2 log ^. log ( - x) = log 2. I.] EXAMPLES. II ./ II. --^dXy J 12. t'dXy -^13. I sin 6^9, •' o x/ 14. COSOT^X, 5- J„ cos' 61' ■ Jo 4/(«'-.v')' 2 4/a:(d!^ + l^jc + ^x"") I = 23!-^ • a'\ f^ — I. I — cos c. sin .r tan/9 = o. irr X~]i-^ 7t sin-^ - 1 = - . >/ 19. If a body is projected vertically upward, its velocity after t units of time is expressed by a denoting the initial velocity ; find the space .fi described in the time U and the greatest height to which the body will rise. \ ^^ =\ 1) dt — at, — -kg^r^y , .a a when V — o ,f= ~~,s = — i^ 2g Kk- i^ 12 ELEMENTARY METHODS OF INTEGRATION, [Ex. L N 20. If the velocity of a pendulum is expressed by nt V =^ a cos — the position corresponding to / = o being taken as origin, find an ex- pression for its position s at the time t, and the extreme positive and negative values of j. 2ra . 7tt s = — — sm — TT 2r s = ± when f=T, ^r, ^t, etc. I 21. Find the area included between the axis of x and a branch of the curve y — sin X. 2. [ 22. Show that the area between the axis of x, the parabola and any ordinate is two thirds of the rectangle whose sides are the ordinate and the corresponding abscissa. A 23. Find (a) the area included by the axes, the curve and the ordinate corresponding to .^c = i, and (/3) the whole area be- tween the curve and axes on the left of the axis oiy. {a) 8 - I, (/?) I. I 24. Find the area between the parabola of the nth degree, and the coordinates of the point (a, a). , » + I § I.] EXAMPLES. 13 ^25. Show that the area between the axis of .r, the rectangular hyperbola xy—i, the ordinate corresponding to :r == i, and any other ordinate is equivalent to the Napierian logarithm of the abscissa of the latter ordinate. For this reason Napierian logarithms are often called hyperbolic logarithms. J 26. Find the whole area between the axes, the curve and the ordinate for x =. a^m and n being positive. li n> m. na n — m if n^m^ C50. / 27. If the ordinate BR of any point B on the circle x^^f^a' be produced so that BR • RP = a"^, prove that the whole area between the locus of R and its asymptotes is double the area of the circle. J 28. Find the whole area between the axis of x and the curve y (a' + x') = a\ 7ta\ 29. Find the area between the axis of x and one branch of the com- panion to the cycloid, the equations of which are xz=zaip y = a (i — costp). 14 ELEMENTARY METHODS OF INTEGRATION. [Art. II. II. Direct Integration. II. In any one of the formulas of Art. lo, we may of course substitute for x and dx any function of x and its differential. For instance, if in formula (b) we put x — am. place of x^ we have J -^~a ^ ^^^ ^^' ~ ^^ ^^ ^^^ ^^ ~ ^^' according as x is greater or less than a. When a given integral is obviously the result of such a sub- stitution in one of the fundamental integrals, or can be made to take this form by the introduction of a constant factor, it is said to be directly integrable. Thus, sin 7;^ ;r ^jr is directly in- tegrable by formula {e) ; for, if in this formula we put mx for B, we have j sin nix • 7ndx = — cos mx , hence sin mx • mdj; = — — cos mx. I sin mx dx — — J /// J So also in ^ 4/(^ 4- b^) x dx , m the quantity x dx becomes the differential of the binomial {a + bx^) when we introduce the constant factor 2/^, hence this integral can be converted into the result obtained by putting (a + b:^^ in place of ;i: in j^ xdx^ which is a case of formula {a), lUS ['^{a^b:^)xdx = —-\{a-V b:^f 2bx dx = -~(a + bx'f . Thus §11.] DIRECT INTEGRATION, 1$ 12. A simple algebraic or trigonometric transformation sometimes suffices to render an expression directly integrable, or to separate it into directly integrable parts. Thus, since — sin X dx is the differential of cos x, we have by formula ifi) f , ( sin X dx , tan X dx = = — log cos x . J J cos ^ So also, by formula (/), ' \x2.n^ ddS ^[{s^ee- I) ^^= tan (9 -6*; by (e) and {a), fsin^ Ode = j(i - cos2 6) sin 6'^6> = - cos ^ + j cos^ ^ : "' \^ by (j) and (a), "^1 ^(1 -;^) -^j^^ -4:^)-^(-2^^^) = sin-^r- 1/(1 -^. Rational Fractions. "^ \ \/7--^ 13. When the coefficient of dx in an integral is a fraction whose terms are rational functions of x^ the integral may gen- erally be separated into parts directly integrable. If the de- nominator is of the first degree, we proceed as in the following example. Given the integral y — ^ ^dx\ ^ J 2;ir + I by division, 2x + 1 2 4 4 2^ -f i' 1 6 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 3c hence dx '^-'*id,=.'-\,d.^nd,*^^ H--y 2x + I 2J 4J 4 J 2;ir + I 4 4 o When the denominator is of higher degree, it is evident that we may, by division, make the integration depend upon that of a fraction in which the degree of the numerator is lower than that of the denominator by at least a unit. We shall consider therefore fractions of this form only. Denominators of the Second Degree. 14. If the denominator is of the second degree, it will (after removing a constant, if necessary) either be the square of an expression of the first degree, or else such a square increased or diminished by a constant. As an example of the first case, let us take The fraction may be decomposed thus : X -V \ X — \ -\- 2 I 2 (x - if ~ (x -if - x-i^ (x- if ' hence [ X + I J [ dx ^ [ dx = log {x - I) (6. The integral f . ^ "^ ^ . dx §11.] DENOMINATORS OF THE SECOND DEGREE. 1/ affords an example of the second case, for the denominator may be written in the form x^ -V 2x -\- 6 = {x -^ \f 4- 5. Decomposing the fraction as in the preceding article, •^ + 3 _ X ^\ 2 ^^X-^ij c^- whence >. ' [ ^ + 3 ^^ , f(^+ ^)dx [ dx ^ The first of the integrals in the second member is directly integrable by formula {b), since the differential of the denom- inator is 2 (;r + \)dx^ and the second is a case of formula (k'). Therefore » I*" (X ■\- X 2 X •\- \ ^ '^ -2 . ^ . ^ ^-y = i log V-^'^ + 2;ir + 6) + -— tan-^ — -— . ;r + 2;f + 6 ^ ^ \ / y^ yg c 16. To illustrate the third case, let us take f 2.r + I \x'- X -6 dxy in which the denominator is equivalent to (x — yf — 6^, and can therefore be resolved into real factors of the first degree. We can then decompose the fraction into fractions having these factors for denominators. Thus, in the present example, as- sume 2;ir + I A B x^ — X —6 X— ^ X + 2 (0 in which A and B are numerical quantities to he determined* Multiplying by {x — 3) (x ~\- 2), 2X+ I =A{x -{- 2) -{- B{x-^. (2) 1 8 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 6. Since equation (2) is an algebraic identity, we may in it assign any value we choose to x. Putting ;tr = 3, we find 7 = 5^, whence A=h putting X — -2, ' - 3 = - 5^, whence B = \.' Substituting ' these values in (I). 2X + I 5(^-3)^ 5(^ 3 ■ .1 ;,^_^_6- • + 2)' / whence f 2X ^- \ ^dx = I log (x - 3) + I log {x + 2). 17. If the denominator, in a case of the kind last considered, is denoted by (x — a) (x — d), a and b are evidently the roots of the equation formed by putting this denominator equal to zero. The cases considered in Art. 14 and Art. 15 are respectively those in which the roots of this equation are equal, and those in which the roots are imaginary. When the roots are real and unequal, if the numerator does not contain x^ the integral can be reduced to the form f dx ]{x-d){x-by and by the method given in the preceding article we find f dx T \x — a) {x — b) log {x - a) - log {x - b)\ '<-^y ^^y * The formulas of this series are collected together at the end of Chapter II., for convenience of reference. See Art. loi. § II.] DENOMIN-A TORS OF THE SECOND -DEGREE. IQ in which, when x < a, log {a — x) should be written in place of log {x — a). [See note on formula (b), Art. lo.] If ^ = — «, this formula becomes f_^ = ±iog^:::f ^a') }x^ — a^ 2a ^ X -{- a ^ ^ Integrals of the special forms given in (A) and (A') may be evaluated by the direct application of these formulas. Thus, given the integral f ^^ }2x^ -{- ^x — 2' if we place the denominator equal to zero, we have the roots a = ^, d = — 2; whence by formula (A), dx 2^ + 3.r — 2 dx _ I I - X — \ ^ (.tr— I) (JT2) ~ 2 ■ 2i °^ X ^ 2 ' or, since log {2x — i) differs from log {x — l) only by a con- stant, we may write f dx I - 2jr — I log 2x^ -{- ^x — 2 5 ^ X + 2 Denominators of Higher Degree. 18. When the denominator is of a degree higher than the second, we may in like manner suppose it resolved into factors corresponding to the roots of the equation formed by placing it equal to zero. The fraction (of which we suppose the numerator to be lower in degree than the denominator) may now be decom- posed into partial fractions. If the roots are all real and un- equal, we assume these partial fractions as in Art. 16 ; there being one assumed fraction for each factor. li, however, a pair of imaginary roots occurs, the factor cor- 20 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 8. responding to the pair is of the form {x — of -\- ^^, and the partial fraction must be assumed in the form Ax ■\- B (x - of + fi :2' for we are only entitled to assume that the numerator of each partial fraction is lower in degree than its denominator (other- wise the given fraction which is the sum of the partial fractions would not have this property). 19. For example, given ^ dx. ][:x^ ^ \){x - i) Assume jr + 3 Ax + B C , . {:>^ •¥ \)(x —\) x^ + I X — 1 whence X + 3 = (x - i){Ax + B) + {x^ + i) a Putting X = ly 4 = 2Cy whence C=2; putting X = Of 5 = — B -h Cy whence B = — i. To determine A, any convenient third value may be given to X ; for example, if we put x = — i, we have 2 = -2{-A + B) -h 2C .-. A=^2, Substituting in (i), ;r+3 _ 2 2;ir+l {x^ -\- i){x— i) ~ ^^^ ~ ;r^ + I ' §11]. DENOMINATORS OF HIGHER DEGREE. 21 therefore J;tr + 3 J _ [ dx {2xdx { dx = 2 log {x — i) — log (ji:^ + I) — tan" ^ X. 20. If the denominator admits of factors which are func- tions of jc^y and the numerator is also a function of or^, we may with advantage first decompose into fractions having these factors for denominators. Thus, given f x^dx ]x*-a*' Putting J/ for x^ in the fraction, we first find hence ^ _ I I f x^dx _ I ( dx C dx therefore [see equation {A'), Art. 17], x^dx _ 4a fx^dx I , X — a I . ;r -T 1 = — log — — - + — tan- ^~ , XT — or Aa X -{• a 2a a This method may sometimes be employed when the nume- rator is not a function of x^ ; thus, since xf'-a'- 2a\x^ - a") 20" {x" + a")' we have x^-a' 20" {x' - a") 2a^ {x^ + a^) ' 22 ELEMENTARY METHODS OF INTEGRATION. [Art. 20. hence X dx I , x^ — c^ log ]x!'-(^ 4^ ^;r^ + ^2• 21. The fraction corresponding to a pair of equal roots, that is, to a factor in the denominator of the form {x,^ dfy is (see Art. 14) equivalent to a pair of fractions of the form A B + X — a (x — a) ■2 ' we may, therefore, at once assume the partial fractions in this form. We proceed in like manner when a higher power of a linear factor occurs. For example, given we assume X + 2 A B ^ C D + 7 ^, + r + (x — if{x + 1) {x — if ' (x — if ' X — I ' X + I whence x-h2=lA-{-B{x- i) + C{x-if]{x+ i)^D{x-if. . (i) Putting ;r = I, we have 3 = 2A .-. A=i. The values of B and C may be determined as follows : if we substitute the value just determined for Ay equation (i), is identically satisfied by x = i, hence it may be divided by jr — i. We thus obtain ^^=[B + C{x-i)-](x+ i)+D{x-if . . (2) § IL] MULTIPLE ROOTS. 23 in which we may again put x = \, whence B — — \. In like manner from (2), we obtain l^C{x-^ \)^D(x-\), from which C —^^ and Z> = — J. Therefore r x^2 J _?>[ d,x \ [ dx \[ dx \ { dx J(;ir-lf(;r+i) 2j(;r-i)«~4J(;r- i)2'^8j;^^~8 J^TTl ^ I \ , X- \ X A(x-\f \(x - i) ' 8 ^;ir+ I 22. In this example, after obtaining the values of A and D from equation (i) by putting ;tr = i, and x = — \^ two equations from which B and C might be obtained by elimination could have been derived by giving to x any two other values. Con- venient equations for determining B and C may also be obtained by putting ;ir = i in two equations successively derived by differentiation from the identical equation (i). In the first dif- ferentiation we may reject all terms containing (x — if \ since these terms, and also those derived from them by the second differentiation, will vanish when x = \, Thus, from equation (i), Art. 21, we obtain \ — A ^ 2Bx + 2C (x^ —i) + terms containing {x — if. Putting X = I, and ^ = | , we have B = — I. Differentiating again and substituting the value of B, o = — ^ -\- 4Cx + terms containing {x — i), and, putting x = i m this last equation, C = \ . 23. When the method of differentiation is applied to a case 24 ELEMENTARY METHODS OF INTEGRATION, [Art. 23. in which more than one multiple root occurs, it is best to pro- ceed with each root separately. Thus given, f -y + I . ](x- \f(x\2f' (x — \f (x -^ 2f (x — if x—i {x + 2f X+2 whence x+i={A+B{x-i)-]{x + 2f-^[C+D{x + 2)]{x-if . . (I) Putting ;r = I, and ;r = — 2, we derive A = '-^ C=-'-. 9 9 Differentiating (i), we have I = 2A (x + 2) + B {x + 2y -^ terms containing {x — i), 2 I whence, putting x = ly and A == - , we have B = . Again, differentiating (i), we have I = 2C {x — i) -i- D (x — if + terms containing (x + 2), whence, putting x = — 2y and C = , we have D = — . Therefore f X -\- I _ _ 2 I J_, X + 2 ]{x - if {x + 2f^ ~ g{x- l)"^ 9 (;ir + 2) "^ 2y^^x- I * 24. Instead of assuming the partial fractions with undeter- §11.] RATIONAL FRACTIONS. 25 mined numerators, it is sometimes possible to proceed more expeditiously as in the following examples : Given U (I + ^) dx\ putting the numerator in the form i + .r^ — ;i:^, we have 'dx_ ,3 x{i + x^) Treating the last integral in like manner, X dx 1? = -^-log^ + Jlog(i+^)=-^+log-ili^. Again, given putting the numerator in the form (i + xf — 2x — x^, we have f I , _ [dx r 2 + X , J;^(i ^xj'^'^-]l?-]x{Y +xf'^'' _ [dx r dx f dx ~J:? " ^]x{i + xf^iii +xY' Hence by equation {A), Art. 17, dx C dx I , + X I + X X 26 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. y, Examples II. <=i^^' /r.f^, -log(«-.). J fdx {a-xf' 7- d - (a' - x')^^ 3 ^a"--^" (a — xf^ a — X [ XdX I 1 / 2 , 2\ / f jv' dx . , I , 8. (« + wj;)" dx^ 9- Jsin"2^' '^^^ V lo. \ co^^ X ?,m.x dx, j'^^.w., ^j.v^' ' 1 f cos dB , >i 12. sec' 3 A- tan 3^ <3^r, •^ - (^mTW^ («^ + 3^y 24 (^ 4-;«a)'-«'' 3»« cot 2;«' 2 I — COS* X I 2 cosec'' 0. sec' 3-^- I 9 §11.] EXAMPLES, 27 / 13. fd!'«-^^, (Km^ ^T • "^ J ^ m\oga >1 14. I (f-^ - \fdx, - -JfS-^ — |f=^ + 3£^ - ^. / f/ , • 8 \9 • ^ . ^ (i + 3 sin' J?)' V 15. (i + 3 sm :r) ^\XiXQ.O'i>xax, \. -^-iL ) ^ ^. J o '^yiax — x^) Jo v| 17. Kos^'o^e, -^^«('---^^'^ ^^ -. ^ 18. I sec* Q ^9, 7'^^^^^^ A^' ^ tan0+-tan'O. 19. tan'^y^, — tan"^ + logcos^. IT n V 20. sec* X tan ^ ^jc, kJU,--*.*-^ - sec* ^ = — . Jo . 4 Jo 4 . [4/ ^ ~ ^ dx, ezsin"^- + 4/(^' — ^'). jy a + X ' a ^ J Z 2 4 |/ ^jf, i^{2ax — x"*) -\- a vers " ^ - . • f . / N ^ cos(«— 29) 24, sin {OL — 20) ^0, ^.SU^^e^y- • 21. 28 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. 25 f CO! cos X dx V ''■ l: ^ sin ^ * dx tan.r * ■7t>«^ — -7 log (/2 — ^ sin J^). i log 2. ^'^' J „ tan .r ' i log 2. r- log (- log.T)j' = -log 4 tan- ^f^. ^ 30. j^. r-^' ^33.}^, .^^i^V ^34. j;:,,,^,?^^, r%^^ ^ v/35. fprj^q- V3 tan" 3 tan - 'x\ I 2 sm ^^. I 4^3 sin" .1 -^4/3 i/5 * I 4/10 r tan" 1 ^Vs 4/2 • 4 ' 2.Jk: + -11' TT zVi " . w.v,^y^ ^11.] EXAMPLES. dx v(s -4^- ^y I ^^y^ ^ / 3.r V"^ 29 cos"^ f . 'V^v. / 38. P'-J^yr, J^^ J o H ~~" AT V 39. ] .,^ + I ^> fJi:' +^ 4- I ^ I 4^- J ^^ _^ 4- i ^'> -*■ + ^^g (-^^ - ^' + i) + VC^'' — a'')— a sec ~ * - . a' (log 2 - t). 4^ — J log (a:' + i) — tan ■ ^ ^'. 2 , 2^1' — I tan"' Vs Vs 41. a: +^log — — 4 ° Jt' + 2 i°g(r^-^--"- /43. Ji^^^^. ^bXi^X' -- -^ + ^ log (^^ + 3). ^ 45- J^., .^.,.^^ '^-^, ■ ■ ^ + Iog X — I 46 r J o dx t,^..JrXC^ X — 2ax cos a -\r a a sm , .V — <3! COS or\ ^ 7C — a — tan ' ^ -. = -. OL ^ Sin ^ 2a sm a — lo 30 ELEMENTARY METHODS OF INTEGRATION. [Ex. 11. t' 47. 48. 49. \/ 50. 1.^ dx 2ax sec « + « y 51- 52. 53. \y 54. / V 55- ^J 56. / M 57- 58. f dx J 2.v' —4^—7' [• x" dx \i-x'" J x" — jt"" — 2J«; ' J (^ + 2) (^ + 3)"' x^^ x" -^ x^ i' J ^^ + ^« _ 2 » f ^' — jr + 2 , J :<:* - 5^' + 4 r ^^ ]x' - x^ - X ^ y' X — a sec «: — ^ tan or 2a tan a ° :f — « sec a -\- a tan a; log ^2 , 2^ — 2 — 3 4/2 12 ^ 2Jt: — 2 + 3 4/2' I , I + ^ 2 log :r + 2 ^ + 3 -[_tan ^^ + log ^^^ J. 7 log — ■ — +-^— tan 6 ^ x-^- \ 3 1/2 • log 2, ^+11- X— 2 -log ; + -log 3 ^X-V2 3 ^X-\ I . Jg + I I 4 °^ — I 2{x — l) 2 (X^ + l)^ I , (^ + l)' , I ^ _i2:t: — I -log-^^ '- — H ;— tan ' — — 6 ^^ — ^ + I 4/3 4/3 1(^^#FT7)' 7iog(— i)-^iog(^' + x)-^). § II.] EXAMPLES. "^■^ log « — - log (i + a;) log (i + a;') tan - ' x. 6'- j J^+^/_6:, '^^' ilogx + ilog(^-2)+ilog(* + 3). / , f x^dx \ ^ X — 2 \/ X X ^ 62. \- T, , -log ; + "^ ^ —-- }x' — X' — 12' J ^ X -h 2 tan- 7 ^ X -h 2 7 V3' J f xVjc I ^ — I ^ ^- ](x'-iY' 4 ^^^r+7" 2(x'^- 1)* i '64. f^4^^., Xtan-^^--i-log^. ^ ( xdx \ . X^ — 2 (/ [" ^^ 7t V J^x'Ca" + xy ' ■ 41 ■ \l [ dx n \^ .11 32 ELEMENTARY METHODS OF INTEGRATION. [Ex. 11. / „ f ^^ \^v. / ,, f ^ ?A-- * "• ]x'{a^bx'y J 74. Find the whole area enclosed by both loops of the curve yj 75. Find the area enclosed between the asymptote corresponding to X = a, and the curve lr»rr _ X 1 I log- i+X ' I +x' i'°s« x' + bx" I , ^ , d? + dx' 2ax^ ^^"'"^ x' ' J \ 2 2 • 2 2 2 2 x" y" + ax — ay. 76. Find the whole area enclosed by the curve a'f = x' {a- - x'). 77. Find the area enclosed by the catenary the axes and any ordinate. £^ -H £ ']■ ^[•-•-•} 78. Find the whole area between the witch .ly = 4a'' {2a — x) and its asymptote. See Ex. 23. 4;ra^ § III.] TRIGONOMETRIC INTEGRALS, 33 III. Trigonometric Integrals. ~ - 25. The transformation, tan'^^ = sec^ ^ — I, suffices to separate all integrals of the form Itan^ede, (I) in which n is an integer, into directly integrable parts. Thus, for example, ftan« ede = [tanS 6 (sec^ 6 - i) dd _ tan^ 6 i ~~4 J ^^"'^ 't^n^Bde, Transforming the last integral in like manner, we have r. 'i n 7n tan* tan^ 6 r . , „ tan^<9^6>=: + tan OdO; hence (see Art. 12) L nn^n tan*<9 tan2(9 , . tan^ Odd = log cos 6. When the value of n in (i) is even, the value of the final inte- gral will be 6, When n is negative, the integral takes the form [cot"^^^', which may be treated in a similar manner. 34 ELEMENTARY METHODS OF INTEGRATION. [Art. 26. 26. Integrals of the form [sec«(9^(9 (2) are readily evaluated when n is an even number, thus [s^eedB = [(tan^ + i)2 sec'ede = [tan* ^ sec2 6 dS + 2 [tan^ ^ st^ddd + [sec^ ^ ^^ tan» 6* 2 tan^ (9 = —r— + ;; + tan 6. 5 3 If ft in expression (2) is odd, the method to be explained in Section VI is required. Integrals of the form cosec*^6d6 are treated in like manner. Cases in which sin'^ Q cos'' Q dd is directly integrable, 27. If n is 2. positive odd number, an integral of the form I sin'^ d cos« Q dB (3) is directly integrable in terms of sin B, Thus, [sin^ B cos^ BdB= [sin^ ^ (i - sin* 6')2cos Bdd _ sin^ B 2 sin^ B sin^ B ~"T~ 5 "^ 7 ' This method is evidently applicable even when m is frac- tional or negative. Thus, putting ;y for sin B, § III.] TRIGONOMETRIC INTEGRALS. 35 ^m«-f^-\>-*"-V'r. ■cos'^ j^_ f(i -J)dy y\ hence •cos^ e ^^ i2i 23 + sin^ e f COS^ ^ 1 2 i 2 Jsint6l -^ r 'h 3-" 3 i/(sin^)- When m in expression (3) is a positive odd number, the in- tegral is evaluated in a similar manner. 28, An integral of the form (3) is also directly integrable when m + n /j an even negative integer^ in other words, when it can be written in the form J cos'«+^^ d J in which q is positive. For example, dd fad f ^-r^ r^ = (tan ^-^ sec* 6 dd sma ^ COS8 ^ J ^ ^ = f(tan (9)-t (tan^ ^ + i) sec^ddd ; hence Jsma = -tant^ ^cos^^ 3 tan«^ It may be more convenient to express the integral in terms of cot 6 and cosec 6, thus i^^i^ = jcot* d (cot2 ^ + I) cosec'ede cot"^ e cot« (9 36 ELEMENTARY METHODS OF INTEGRATION. [Art. 28. Integrals of the forms treated in Art. 25 and Art. 26 are in- cluded in the general form (3), Art. 27. Except in the cases already considered, and in the special cases given below, the method of reduction given in Section VI is required in the evaluation of integrals of this form. The Integrals sln^ e dd, and cos^ d dd. 29. These integrals are readily evaluated by means of the transformations sin^ 6 = ^{i — cos 2d), and cos^ ^^ = i(^ + cos 26), Thus [ sm^Odd = ^{dd-'\ [cos 2ddd = ^d -ism 28, or, since sin 26 = 2 sin 6 cos ^, [sin2 dd6 = i(d - sin 8 cos d) {B) In like manner \cos^ d do = ^{6 + sm 6 cos 6) {C) Since sin^ 6 + cos^ 6 = i, the sum of these integrals is \d6\ ac- cordingly we find the sum of their values to be 6. In the applications of the Integral Calculus, these integrals frequently occur with the limits o and ^n ; from {B) and {C) we derive IT TI f'sin2^^^=|'cos2^^^ = i;r. §111.] TRTGO.VOME TRIG IN TEGRA LS. 37 The Integrals \^^^^, j^^, and J^. 30. We have f de [speeds , , . ,^, ^— 5 2) = —^ — z~ = log tan 6. , . , (D) J sin 6 cos 6 J tan ^ ^ ^ ^ Again, using the transformation, sin ^ = 2 sin ^6 cos J^, we have Jsin6/~Jsini^cosi^~J tan J^ ' hence |^,= logta„ift (E) This integral may also be evaluated thus, de [ smddd f sin6>^^ f d6 _[• sm ddd [2 Jsin^~J sin^^ '"U cos^ 6 * Since sin Odd = — d{cos 6), the value of the last integral is, by formula {A'), Art. 17, I , I — cos ^ , /I — cos B I , I — cos c/ I /I — cos^* and, multiplying both terms of the fraction by I — cos 6, we have { dS , I — cos 6 , r^,. 38 ELEMENTARY METHODS OF INTEGRATION, [Art. 3 1. 31. Since cos 6 — sin {^n + ^), we derive from formula (E), J cos<9 Jsm(i;r + <9j ^ L4 2J ^ ^ By employing a process similar to that used in deriving for- mula {E), we have also [ dS . I + sin <9 Miscellaneous Trigonometric Integrals, 32. A trigonometric integral may sometimes be reduced, by means of the formulas for trigonometric transformation, to one of the forms integrated in the preceding articles. For example, let us take the integral de f ^ J ^ sin ^ + <^ cos d* Putting a^kQosa, b — k 'saw a, , . . . (i) we have [• dS i_ f dB J k ^ 2^ ^ " or, since equations (i) give ;^=. V(^2 + ^), tan« = -, f J^ , = - , ' ,, log tan i fe + tan- ■:*1 . J a sin 6* + i^ cos 6* */(a^ ^ I?) ^ 2L «J § III.] MISCELLANEOUS TRIGONOMETRIC INTEGRALS. 39 33. The expression sin ntd sin nQ dd may be integrated by- means of the formula * cos {m — n) 6 — cos {m + n) 6 — 2 sm md sin nd ; whence \smmdsmnddd = — ) { } f- . . (i) J 2{m — n) 2 {in -{- n) ^ ' In like manner, from cos (m — n) 6 -{■ cos {m + n) 6 = 2 cos mO cos nO, we derive f X. /. //, sin (m — 71) 6 sin (m -V n)d , . \cosmecosn6dd = \ ^+ ) (—. . (2) J 2{m — 11) 2 {m + n) When m = fty the first term of the second member of each of these equations takes an indeterminate form. Evaluating this term, we have sm^nddd = , (3) J f 2 /I -7/1 ^ sin 2nO f n and \q.o%^ nd dS = - ■{ (4) J 24/2 Using the limits o and n we have, from (i) and (2), zdlan. m and n are unequal integers^ sin w^sin;^^^^ = cos mS cos 716 dd = O', .. . (5) Jo Jo but, when in and n are equal integers, we have from: (3;) and (4) [ sm^ nddd = [cos'' ndde =- . . .. . ,. (6) Jo Jo" 2 34. To integrate 4/(1 4- cos 6) dB; we use the formula 2 cos^ ^6 = L -h cos. 6, 40 ELEMENTARY METHODS OF INTEGRATION. [Art. 34. whence i^(i + cos ^) = ± V2cosi^, in which the positive sign is to be taken, provided the value of B is between o and n. Supposing this to be the case, we have ' [ V (I + cos (9) dQ = ^/2 [cos \ede = 24/2 sin \^. For example, we have the definite integral It [^ 4/(1 + cos d)dQ — 24/2 sin- = 2. Jo 4 Intezration of 7 -r.* ^ -^ a + cos t/ 35. By means of the formulas I =cos2{,^ + sinH^ and cos ^ = cosH^ - sin^^^, we have f dd _ f d e Jrt + ^ cos ^ ~ J (^ + b) cos2 4/9 + {a - b) sin^ ^8' Multiplying numerator and denominator by sec^^^, this be* comes f sec^idd e ]a-h b + {a-b)t3in'ie' and, putting for abbreviation tan ^6 = y, we have, since | sec^ 1-6 d6 = dy, [_ de_^ ^ 2 [ dy J^ + 1^ cos 6* ]a ■\- b ^{a - b)f' §111.] MISCELLANEOUS TRIGONOMETRIC INTEGRALS. 4I The form of this integral depends upon the relative values of a and b. Assuming a to be positive, if b^ which may be either positive or negative, is numerically less than a^ we may put a — The integral may then be written in the form 2 f dy a-bjc" + f' the value of which is, by formula {k')y c{a — b) c Hence, substituting their values for y and c, we have, in this case, f— 4^=-7-^-^tan-4V'^t^^4^]- . (Q ]a-^ b cos Q ^(a^ — 1^) \J a -V b ^ J If, on the other hand, b is numerically greater than a, this expression ior the integral involves imaginary quantities ; but putting b ^ a _ « the integral becomes dy _ f dy , b the value of which is, by formula .(^'), Art. 17, c{b — d) ^c-y ) 42 ELEMENTARY METHODS OF INTEGRATION. [Art. 35. Therefore, in this case, f de _ I sf(b^d)-^W{b-d) tan \Q 36. U e < I, formula (G) of the preceding article gives f 23= -7^^— ^tan-^r^/^^tanl /9 . . (i) J I + ^ cos <9 V {i — e^) [_y 1 + e ^ J ^ ^ Putting |/^^-tani^=tani0, (2) and noticing that # = o when ^ = o, we may write Now, if in equation (i) we put ^ for 6 and change the sign of Cf we obtain f —J^ = —A^ tan- [^/'L±^ tan i ^1 ; J^ I — ^ cos ^ V{i - e^) [Jy I — e ^ ^J' hence, by equation (2), f ^^ - ^ ■ /,^ J^i-^cos^~ i/(i-^^) ^^^ Equations (3) and (4) are equivalent to de ^ d^ i+^cos^ |/(i-^)' ^5^ , d^ de .^. and T = -77 2^ , (o) § III.] . TRIGONOMETRIC INTEGRALS. 43 the product of which gives (i + ^ cos ^) (i — ^ cos ^) = I — ^ . . . . (7) By means of these relations any expression of the form f dS J(i + e zo^ey where ;2 is a positive integer, may be reduced to an integrable form. For f dS f de I . J(i + ^ cos^)'^ ~ Ji + ^ cos6> (i + ^ cos(?)«-^ * hence, by equations (5) and (7), dd e cos Jji + ^cos^)«-(i-^)«-*J,^^ By expanding (\ — e cos ^)'*~% the last expression is reduced to a series of integrals involving powers of cos ^ ; these may be evaluated by the methods given in this section and Section VI, and the results expressed in terms of d by means of equa- tion (2) or of equation (7). Examples III. . , tan^ mx tan mx tan mx ax, — 1- x. ^m m tan' xdxj 1^ ~ i log 2. y^. fsec* (6 + a) de, *^'^" "^ "^ + tan (6 + a). 44 ELEMENTARY METHODS OF INTEGRATION. [Ex. III. / r- ^ Jo vs. sin' 6 cos' ^9, "/ 6. |/(sin e) cos' 9 ^0, tr \/ 7. I COS* sin' ^0, , / f sin' dTo 5. i / f ^ 2.3. 4.3: , 2 . il ^ - sin^0 — - sin2 B -\ sm ^ 0. ^ 7 II 2 35 f dB 9. ^^ ^— , Multiply by sin' + cos' 0. tan — cot 0. J sin cos / (sm X , T- aXy cos X 10. — r-^/jtr. See Art. 2Z. tan* jr ^'•IS^T^' i(tan'e-cot'e) + alogtane. 1 I.. (i^(!'^, Itante. ' J C0S2 J f sin'jc dx ^^' J cos''^ ' J f sin'jv //jc '4- J-WIT' 5 cos' ^ 3 cos' X tan' a: tan' x 5 3 j 15. |sin'0cos'0A 3V [29 -sin 20 cos 20]. § III.] 7 ^ V i6. 'siri mx dx, EXAMPLES. 45 n 2fn v/ '^ J sm e m COS0 ' / ,8. pS^t^, Ji: sine ' 3 •I Sin + cos 9 + cos X ' log tan — I — — sin ^. i(log3-i). V/ 20. j- f ^JC J 21. , ^ J ^ I — COS X tan \x. I — cot \x. ^ -■ It dx ± sin jc ' Multiply both terms of the fraction ^ i =F sin x. tan x ± sec jp. vi ^*^* JsecQ ± tan 9' log tan - + - L4 2_ ± log cos 9. Nl 24. cos cos 39 ^9. ■ See Art, 33. J sin 4S + i sin 29. IT \/ 25. ^ cos 9 COS 29 rtTs, •'0 I 3* J 26. rsin'9 sin 20 //9, It isin*9l'=i —^0 \/ 27. ^ sin 39 sin 29 d% ' 2 5' 46 ELEMENTARY METHODS OF INTEGRATION. [Ex. III. \\ 28. sin m{} cos «0 ^0, I — cos (m -\- n)e I — cos (m — n) B 2 {m -\- n) 2 {m — n) I 29. cos X cos 2x cos 3jr ^JC, Reduce products to sums by means of equation (2), Art. 33. I Fsin Q>x sin aa: sin 2x "1 4L 6 4 2 J |/ (l — COS:r)^;xr, 2 V'2. 1 31. ^ -i ^ ri ' -2 > — tan-M -tan^ . \1 "^ } a cos^ X + i? sm X ab \_a J \ f ^jt: I , tannic 4 ^2. i— , -r- tan" ^ -^ J I + cos' X ' "'Z 2^- J d' cos' ^ - ^' sin' x' 2ab ^^^ 4/2 4/2 d; + ^ tan 9 sin X dx V (3 cos' a: 4- 4 sin' ^) sin x cos' ^ dJ^c 2 «/^ ^ a — b tan Q COS"* J^ cos^^. , f sm jg COS X di >4 35- J J _j_ ^2 CQs'^ X r y^ Putting y /- b \S 40. J 5 + 3 COS do cos 2 COS G — I 2 — tan f B i tan"* [Han J 6]. |-tan-*i3tan|Q{. J_wL-^^3_tani^ ♦^3 ^ I + 4/3 tan i • 3 — cos tan"* V2 COS 9 ^0 2 4/3* ^ cosOj' 6V^ Arl. 36. -; cos , e + cos 9 sm 9 ^- rfTT (i _ ^2^^ ^ + ^cosQ I — r I + ^cos9 (2 + /) TT (i + ^cos9)' 2(1 -.')i 48 ELEMENTARY METHODS OF INTEGRATION. [Ex. III.. f p COS X -\- q^WiX . 45- \ 7\-' — dx, ^ J a cos :v + /? sm jp Solution : — By adding and subtracting an undetermined constant, the fraction may be written in the form p cos X + ff sin X -h A (a cos ^ + /^ sin ^) rj~- — — ~ ^» a cos :<[: + /? sm jc we may now assume / cos X -^ q sin X + A {a cos :r + ^ sin .^t:) = ^ (<^ cos x — a sin x) ; the expression is then readily integrated, and A and k so determined as to make the equation last written an identity. The result is f /» cos ^ + ^r sin Jic , ap -^ bq bp — aq . . , ? \ — ^^ — dx = ^. . ,o X + 3 , ,■; log (a cos ^ + <^ sm x). J a cos X -\- bsmx a^ ^- b' a^ + b' ° ^ ' 46. — ; — T— , See Ex. At^. ax ^ b . f , 7 ■ \ rr> + a I z2 log (a; cos ^ + ^ Sm X). a' + b' ' a' -]- b' 47. Find the area of the ellipse X = a cos ^ ji^ = (^ sin ^. — 4ab\ o sin^ dfdS = nab. 48. Find the area of the cycloid jp = d! (^ — sin ^) jF = <^ (i — cos ^). (27r (i - COS tpY dip = 3a'7r. o §111.] EXAMPLES. 49 49. Find the area of the trochoid (b < a) X = aip — dsintp y = a — d cos ^'. 50. Find the area of the loop, and also the area between the curve and the asymptote, in the case of the strophoid whose polar equation is r =^ a (sec ± tan 0). Solution : — Using as an auxiliary variable, we have / . • \ n , sin'^Q"! ji- = ^ (i ± sin G) y = « tan Q ± L ^ ' "^ L cos Qj the upper sign corresponding to the infinite branch, and the lower to the loop. Hence, for the half areas we obtain + a" [ "sin BdB -^ a'[ sin' G ^0 = ^H i + - and — «' I sin B dB + d^ f sin' B dB = a^\ i . Ji. )^ L. 4J . 50 METHODS OF INTEGRATION, [Art. 37. CHAPTER II. Methods of Integration — Continued. IV. Integration by Change of Independent Variable, 37. If X is the independent variable used in expressing an integral, and y is any function of x, the integral may be ex- pressed in terms of j, by substituting for x and dx their values in terms of y and dy. By properly assuming the function j, the integral may frequently be made to take a directly integra- ble form. For example, the integral [ X dx J {ax + bf will obviously be simplified by assuming y — ax + b for the new independent variable. This assumption gives X = - , whence dx — — a a substituting, we have X dx I J {ax + b)^ ~ a" J \{y-b)dy f I iog7 + ^^; § IV.] CHANGE OF INDEPENDENT VARIABLE. 5 1 or replacing/ by x in the result, [ X dx I 1 / . A\ , ^ 38. Again, if in the integral f dx Jf'- I we put y — e% whence X = log 7, and dx = — , we have f dx _ r dy Jf"— I "~ J j(j— I)* Hence, by formula (A), Art. 17, It is easily seen that, by this change of independent variable, any integral in which the coefficient of dx is a rational func- tion of £% may be transformed into one in which the coefficient of ^ is a rational function of 7. Transformation of Trigonoinetric Forms. 39. When in a trigonometric integral the coefficient of dB is a rational function of tan ^, the integral will take a rational algebraic form if we put dx tan 6 = Xy whence dd = i+;r2 52 METHODS OF INTEGRATION. [Art. 39. For example, by this transformation, we have f de _ f dx Ji -4- tan6'~ J(i +;t^)(i +;ir)* Decomposing the fraction in the latter integral, we have f ^^ _ \{ dx If X dx ^\{ dx J I -h tan d~~ 2)1 -^ x^ 2 J I + x^ ' 2JI + X = ^ tan"\' — i log (i + x^) 4- i log(i + x) '''■ \ i + tan ^ = i C^ + ^^S (^°' ^ + ^^" ^)3- 40. The method given in the preceding article may be em- ployed when the coefficient of d6 is a /lomogeneous rational func- tion of sin Q and cos 6, of a degree indicated by an even integer ; for such a function is a rational function of tan Q. It may also be noticed that, when the coefficient of dd is any rational func- tion of sin S and cos l9, the integral becomes rational and alge- braic if we put e for this gives 2Z sin e I + c^' ^ =:= tan ^ ; cos ^ = ^^^3, This transformation has in fact been already employed in the integration of . See Art. x^, ^ a + b Qos 6 ^^ . § IV.] LIMITS OF THE TRANSFORMED INTEGRAL. 53 Limits of the Transformed Integral. 41. When a definite integral is transformed by a change of independent variable, it is necessary to make a corresponding change in the Hmits. If, for example, in the integral r dx we put X — a tan ^, whence dx — a sec^O dO, we must at the same time replace the limits a and oo , which are values of x, by ^Tt and^;r, the corresponding values of d. Thus L(^F^.=,-J-I:cos^^ dd ~ 2a^L 6 -h sin 6 cos 6 7t — 2 The Reciprocal of x taken as the New Independent Variable. 42. In the case of fractional integrals, it is sometimes use- ful to take the reciprocal of x as the new independent variable. For example, let the given integral be dx }x^{x -h if Putting ^ = -y whence dx = ^ , y y 54 METHODS OF INTEGRATION. [Art. 42. we have y \ y. Transforming again by putting z — y ■\- I, the integral be- comes = _ _ + 3^ _ 3 log ^ - - Therefore, since ^ = r + i = - + i = X X dx_ _ _ (x^ \f 3( -^'+ f . 1 „^ X -^ I (,r + 1)2 ~ 2x^ X X ■\- \ ^ ^ X A Power of x taken as the New Independent Variable. 4-3. The transformation of an integral by the assumption, y^x'^ (i) is not generally useful, since the substitution - I -- 1 X = r«, whence dx — - y"" dy, 11 will usually introduce radicals. Exceptional cases, however, § IV.] THE EMPLOYMENT OF POWERS OF X. 55 occur. For, since logarithmic differentiation of equation (i) gives — -= — , (2) X ny it is evident that, if the expression to be integrated is the product of — and a function of x^ , the transformed expression will be the product of — and the like function of y. For example, the expression {x!" - i) dx x{x^ ^ I) ' dx which is the product of — and a rational function of ;r*, becomes dy , Ay{y^ I) a rational function of y. Hence, decomposing the fraction in the latter expression, we have J ^(^+ I) a) yiy^i) ^ 4 ^ y V (x^ + i) 44. When this method is applied to an integral whose form at the same time suggests the employment of the reciprocal, as in Art. 42, we may at once assume y = x~'^, ^ Thus, given the integral [" dx 56 METHODS OF INTEGRATION. [Art. 44. putting y = x^, whence dx _ dy ~^ ~ " sy' we obtain - _ir ydy 3J12J/+ I = - y , log {2y + 6"^ 12 i)^ °_2-log3 12 45. The same mode of transforming may be employed to dx simplify the coefficient of — , when this coefficient is not a rational function of x^. Thus, the integral r dx }xV{x^-a^) will take the form of the fundamental integral (/'), if we put 09 . dx 2 dy x^ — Ti whence — — _.^i-. X I y Making the substitutions, we have dx 2[ dy 2 -X y 2 _i (x\ 5 — ' -^ — sec -4- = — -, sec ' ^ f ^^ — ? f ^y ]xV(^-ci^) " 3J yV{7~^~^) " ^J ^^^ J " 3^1 \a Examples IV. ^ I- j l^^^'^'> log (2 + '^') + ^ I §iv.] EXAMPLES. 57 / . ' xdx J(i-^r' ^3. f^'--^+'^r J(2^ + ir ' v/4. r .v' p £2^ ^ 7- J -co f- + I ' 8. f '"+' ^X Jl-6-'^-^' ' 2 + tan 6 9- . 3 — tan e ' r ^6 J tan' 6 - i' ' tan' ^0 tan' 6 — I ' cos d^ a cos 6 — ^ sin Q ' 2^—1 2(1- xf- 2A -f I _ log \2X +1) 7 8 "1 8(2;*:+ i) log;/ + 4y — 2 ]-log.-i, ^— (log I -f f-*^). 2 °^ f ^ + I I — log 2, €-»^ + 2 log (f-* — l). e — log (3 COS — sin e) I , tan 0—1 —log 4 ° tan + 1 2 I , tan — 1,0 - log + - , 4 ^ tan + I 2 aB — b log (a cos — <^ sin 0) a' + b' 58 METHODS OF INTEGRATION, [Ex. IV. f COS 6 ^& r» ^ .' •^ Jcos(a' + ''^' (9 + oc) cos « — sin « log cos (9 + a). [ sin (0 '+ a) (6 + /5) cos {a— ft) + sin (or — /?) log sin (9 + )5). 5. tan (9 + «:) cos 9 ^/9, —cos 9 + sin <^ log tan 2B + 2a -\- 7t , {'^ COS B dB , . . , . ' Jo sin (a: + 9) ^ cos « log (2 cos Of) + Of sm^. IT IT rr cos j 9 ^^ I ^ 1^2+2 sin9 n6 _ log (3 + 2 ^2) Jo COS 9 ' 4/2 ° ^2 — 2 sin 9 Jo 4^2 _ fsin|9^/9 , ^ 18. ^^-r— , log tan J sin9 ^ TT + fx' dx /z' 20. l^'(l+-v')' '°^ r* jtr'^jT I fs Jo (l + -^T' 4 Jo 4/(1 + x") _I_^ a; 2ar'' 21. I 7-— T-— iTs, - I ~ sin" 2B dB— —r § IV.] EXAMPLES, 59 n Jdx L 4- £ _ 1 X -\- \ X [ dx I I , , X ^^- J x(^'-i) ' flog(^'-.)-logx. o _ 2 - -log 3 I 8 I ;'°s^ x' 4^ + ^.rV* I Inor — Jt:^ ^n &^« + ^a' e/jf 2 _ /^ V. Integrals Containing Radicals, 46. An integral containing a single radical, in which the expression under the radical sign i s of the first degree, is rationalized, that is, transformed into a rational integral, by- taking the radical as the value of the new independent vari- able. Thus, given the integral f dx 14- v{x-\-\y 60 METHODS OF INTEGRATION, [Art. 46. putting J = V(^ + I), whence X — f — I, and dx = 27 cfy\ we have f ^- J I + 4/(;r + -Ay^y = 2\dy 2f '^y i) Ji +7 J "^ Ji +j/ = 2j-2log(l +/) = 2^/(x + l) - 2log[l + »/{x + l)]. 47. The same method evidently applies whenever all the radicals which occur in the integral are powers of a single radical, in which the expression under the radical sign is linear . Thus, in the integral dx ,(^-_i)5 + (^--i) ± » the radicals are powers of (;ir — i)^ ; hence we put y = {x — i)K and obtain dx ^^f^ fdy = 6\\y- i)dy + 6f -f^= -3 +6\og2, Jo Jo J + I 48. An integral in which a binomial expression occurs under the radical sign can sometimes be reduced to the form considered above by the method of Art. 43. For example, since f dx ix{x^-^ i)^ § v.] INTEGRALS CONTAINING RADICALS. 6l fulfils the condition given in Art. 43, when n — 3, the quantity under the radical sign may be reduced to the first degree. Hence, in accordance with Art. 46, we may take the radical as the value of the new independent variable. Thus, putting whence ^ =^ ^ — i, and we have dx 4js'^ dz X -3(^-1)' { dx _4 fj^ dz ixix^ + i)i~3 J^-i* Decomposing the fr^^ction in the latter integral as in Art. 20, we have finally _^ — _ tan'M (;i^ + I) + - log^- \ — - . Radicals of the Form V{ax^ + 3). 49. It is evident that the method given in the preceding article is applicable to all integrals of the general form. \x'^'''-^'{ax^ i- dy+^dx, (I) in which m and 7i are positive or negative integers. These integrals are therefore rationalized by putting y = V[a^ + b).. 62 METHODS OF INTEGRATION, [Art. 49. Putting m = O, the form (i) includes the directly integrable case f(^,i^ + by + * xdx. 50. As an illustration let us take the integral dx f.7 X V{x^ + a^) ' putting J r= V{^ + a^), , 2 5 2 1 ^^ y ^y whence x^=t — ^ , and — = ■ / ^ , , X y — a'^ we have dx _ r 3 and dx = -%, we obtain f dx _ r y dy J (ax" + bf ~ ~ J '{a-\-bff The resulting expression is in this case directly integrable. Thus [ ^^ ^ ^ _ ^ / 21/^ + 2 log (i - Vx), J-a^ ^ 7 4 Jo 2d> (dx 2 _ /2^ — a — T/ 2\ > - tan ' 1/ - jc V(2ax — a) a ^ a Jo^ 9 7 5 Jy- 315 ^ J2:xr? — ^* 4 4 8 v.] EXAMPLES. 71 .10. r(.+ x)t.^., ^'_3/-|^=^3^^^ Jo o 5 Ji 10 40 ^^- 1 ^(7-% ^-^^ 2 (1+ ^) V(l - at). _ Jxdx T, • ,. -7^ r\ » Rattonahze the denominator. X' + f^'* - a'Y Za^ / f dx^ 2 (.y + g)« - 2 (^ + 3)^ V 13. J ^(^ + ^) + ^(^ + ^) » 3 (« - ^) / [ V{x^-VT)dx V{x* + ^) , ^ Inr ^^^^' +0-1 ; ^' J ^ ' 2 +4^°^ v(;t;^4-I)^-I• "J.(?+^' J-"'-?], 'b"^'- X «^{x^ — c^\ — CL sec ~ ' - . etc 72 METHODS OF INTEGRATION. [Ex. V. r x'dx ' ~ ^/ .a , — ^d^\ See formulas (Z) and {K). jXi/(x' -i-a')-- a' log [x + i/^ + a')] 20. — ^^ ^ dx^ a log ^^ + 4/(a — A^ ). —, V(^r' + «') + -log \x + 4/(-a:' + «')] - ^ 2^ 2 2^ log [ |/^:v' + a') + .rj ^ ^ ^ . \ X I C X dx T §V.] EXAMPLES, 73 25. ^{ax" + b) dx, [a >o] Put V(ax^ ■{■ b) = z —x i/a. -^ log \_x s/a + ^(ax" + ^)] + - ^ '^(ax^ ^b). j 2 Va 2 - I Xq ^ + '*^(-^' + ^') 4- « - Via" + ^') V{d'+b') ^x + V{^" + b')-^ a + V{d' + b') 27 f dx Vii +x') ^^' \ixWii-x') ' G; cot H = Vs- 29 V{^'-a') i dx ^V(^'-i)' ^^' Jo (> + «'] I 2 P 2A;' 2 log tan \ \ . 74 METHODS OF INTEGRATION. [Ex. V. ZZ- 3^ (dx I J ::c4/2 ^ ' Jo 4/(^-^0 L Jo V(cix-x')_\ 2 f ^.V 38. 3,4 V ^^^^ 39. Y(2(ix ~ x^)'dx, 40. )/(2ax — x^)-x dx, a^ ^ cos'' (i 4- sin 6) .'/Q = d;' . 2 41. ^{2ax — x'^)'X^ dx, a' j° ^ cos' (i 4- sine)' ^0 = a' p^ - -~] . ^ = 2r = sec I 4 sine)VQ — T6" — X X 2X^ dn 4 § v.] EXAMPLES, 75 j- dx_ J ^y2ax + x^) 42. ./ , .,:iv > by Art. 56, log \x -\- a ^ */(2ax + .r')] + C; ^j; ^r/. 58, log [ »^x -r i/(2d; + a)] + C". ,J{2ax + .r') ' ■^^'"'"^" "^ ■^'^ ""^ '°^ [-t- + « + V(2ax + ^■•')]. ,,. [/_z_^.vr=[ ' ---^-^- . 1, ^^ J '^ 2^ — .r |_ J 1/(2^20: — .r') J tf sin- V{2ax — Jt-'). 45. j ^(3+^l^_^^) ' ^i' ^-^- 56, sin- ^ :^^ + C; ^^ Art. 58, 2 sin-^ ^ "^4^ + ^'• 46. -77 3T, 2sin-H/- =.-;r. 47- JV(3 + ,^_^')'3 49- £V^'^' logCs + ^l'^ sin- . ^~^ - (-^ + 3) ^(3 + 2-^ - ■^'') 2 2 "7^ METHODS OF INTEGRATION. [Ex. V, J __- ^ 50. Find the area included by the rectangular hyperbola y^ = 2ax + x^^ and the double ordinate of the point for which x = 2a. al6V2 — log (3 + 2 V2)]. Find the area included between the cissoid X {x^ + /) = 2ay^ I 51. Fi and the coordinates of the point {a, a) ; also the whole area between the curve and its asymptote. J (— 7t — 2 W^, and ^na^. 52. Find the area of the loop of the strophoid x{x' +/) + a{x' -f) = o; also the area between the curve and its asymptote. , and 20^ ii -^ j /jr -U jQ For the loop put y z=z — x ^ -j- , since x is negative between the limits — a and o. yj 53. Show that the area of the segment of an ellipse between the X minor axis and any double ordinate is ab ^\Vi.-^ — V xy. § VI.( INTEGRATION BY PARTS, 77 VI. Integration by Parts. __ 59. Let u and v be any two functions of x ; then since d (uv) = udv + V duy udv -i-lv dUy uv whence \udv=uv—\vdu (i) By means of this formula, the integration of an expression of the form udVy in which dv is the differential of a known function v^ may be made to depend upon the integration of the expression v du. For example, if u - we have = cos-';ir and I iv = dxy du = dx hence, by equatioi Vii- ^y COS" ' X'dx = ;r cos-^;i: + - xdx '(I -A in which the new integral is directly integrable ; therefore cos-^;r-^;ir = x Q.o?>~^ x ^ 4/(1 — x^'). The employment of this formula is called integration by parts. 7.8 METHODS OF INTEGRATION. [Art. 60. Geometrical Illustration, 60. The formula for integration by parts may be geomet- rically illustrated as follows. Assum- ing rectangular axes, let the curve be constructed in which the abscissa and ordinate of each point are correspond- ing values of v and u^ and let this curve cut one of the axes in B. From any point P of this curve draw PR and PS^ perpendicular to the axes. Now the area PBOR is a value of the indefinite integral u dv, and in like manner the area PBS is a value of \vdu', and we have Area PBOR = Rectangle PSOR - Avesi PBS; therefore ludv = uv — \v du. Applications, 61. In general there will be more than one possible method of selecting the factors u and dv. The latter of course in- cludes the factor dx, but it will generally be advisable to in- clude in it any other factors which permit the direct integra- tion of dv. After selecting the factors, it will be found con- venient at once to write the product u-v, separating the factors by a period ; this will serve as a guide in forming the product § VI.] INTEGRATION BY PARTS. 79 V duy which is to be written under the integral sign. Thus, let the given integral be Lir^ log X dx. Taking :!i^ dx as the value of dv, since we can integrate this expression directly, we have , dx x^ — 3 3J -^ x'^ log X dx = log X' — x^ = —x^ lo£r X x^ dx 3 "^ 3J x^ = -{3^ogx- I). 62. The form of the new integral may be such that a second application of the formula is required before a directly integrable form is produced. For example, let the given mtegral be jt^ cos X dx. In this case we take cos x dx= dv; so that having x^ = u, the new integral will contain a lower power of x: thus x^ cos X dx = x^'s'm X — 2 Lr sin ;r dx. Making a second application of the formula, we have Lit* cos xdx = x^s'mx— 2 x{- cos x) + cos xdx = jr'sin X + 2x cos x — 2s\x\ x. 8o METHODS OF INTEGRATION. [Art. 63. 63. The method of integration by parts is sometimes employed with advantage, even when the new integral is no simpler than the given one ; for, in the process of successive applications of the formula, the original integral may be repro- duced, as in' the following example: e''^^ sin (nx +a)dx = ,,„. . - c°s (^^ + a) ^ ^ U.^ ^^3 (^^ ^ „) ^^ n n] ^ ' — V L 4. _ e^^ V 1 ^„,x sin (fix + a) dx, n n n ft ] ' in which the integral in the second member is identical with the given integral ; hence, transposing and dividing, (^mx ^mx sii^ (ji^ ^ ^) ^^ _ — g— — 2 \in sin {nx + a) — n cos (nx + a)]. 64. In some cases it is necessary to employ some other mode of transformation, in connection with the method of parts. For example, given the integral [sec^^^^; taking dv = sqc^ 6 dd, we have jsec8^^^ = sec^.tan^- jsec^tan2 6'^6''. . . (l) §VI.] FORMULAS OF REDUCTION, 8 1 If now we apply the method of parts to the new integral, by putting sec d tan 6 dS = dvy the original integral will indeed be reproduced in the second member ; but it will disappear from the equation, the result being an identity. If, however, in equation (i), we transform the final integral by means of the equation tan^ 6 = sec^ — i, we have [sec^ ddd = sec ^ tan ^ - jsec* d dO + [sec Odd. Transposing, f % a jn sin 6 f dd 2 sec^ d dd = — 2-^ 4- -; J cos^ J cos ^ hence, by formula {F), Art. 31, fo a ,n sin ^ I , ^ Vn 6^~| sec^ Odd = 2-^ + - log tan - + - . 2 cos^ 62^ L4 2 J 65. It frequently happens that the new integral introduced by applying the method of parts differs from the given integral only in the values of certain constants. If these constants are expressed algebraically, the formula expressing the first trans- formation is adapted to the successive transformations of the new integrals introduced, and is called a formula of reduction. 82 METHODS OF INTEGRATION. [Art. 65. For example, applying the method of parts to the integral we have x"" e^"^ dx = x"" \x''-'€^^dx, . . . . (i) in which the new integral is of the same form as the given one, the exponent of x being decreased by unity. Equation (i) is therefore a formula of reduction for this function. Sup- posing /^ to be a positive integer, we shall finally arrive at the 8''-'^ dXf whose value is — . Thus, by successive appli- cation of equation (i) we have I X" £«^ dx = — n - X"- a Reduction of Ism"' 6 dS and [cos'" d dO. 66. To obtain a formula of reduction, it is sometimes neces- sary to make a further transformation of the equation obtained by the method of parts. Thus, for the integral [sin"'^^<9, the method of parts gives [ sin'«^^6'= sin'«-^(9(-cos^) + (;;^ — OJsin'^-^ ^cos^ <9^^. § VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 83 Substituting in the latter integral I — sin- 6 for cos^ 6, jsin'« edd—— sin"'' -^ 6^ cos (9 + {fn - i) I sin"'- == 6* ^6* - {pi - i) jsin'" 6'<^^; transposing and dividing, we have f sin- ede=^ ^'""'- ^ '^"^ ^ + ^^^^ fsin"-^ ede, . . . (d J m ml a formula of reduction in which the exponent of sin B is dimin- ished two units. By successive application of this formula, we have, for example : [ • (, n in sin' B cos 6 ^ [ . » n m sm^ddd= g + ^\ sm^Odd sin'/9cos 5 sin^/9cos 6 5 3 f . o ^ ,/. — :: — jr • + ^ - Sm^ 6 dd b 64 64J _ sin^ 6* cos <9 5sin^<9cos<9 5.3sin(9cos^ 5*3'^/? 6 6-4 6-4-2 6-4-2 67 By a process similar to that employed in deriving equation (i), or simply by putting 6 = ^7t — 6' m that equa* tion, we find f „ ,^ cos"'-' ^ sin ^ m — I [ ^„ ^ n jn / \ cos'" 6 dd=. + cos''' -^6dd, , . (2) J ;;/ ;;/ J a formula of reduction, when in is positive. 84 METHODS OF INTEGRATION, [Art. 6%. 68. It should be noticed that, when m is negative, equation (i) Art. 66 is not a formula of reduction, because the exponent in the new integral is in that case numerically greater than the exponent in the given integral. But, if we now regard the integral in the second member as the given one, the equation is readily converted into a formula of reduction. Thus, put- ting — n for the negative exponent m — 2, whence m = — n + 2y transposing and dividing, equation (i) becomes f dO cos^ n — 2 f dS , . Jsin«(9 (/2— i)sin''- Again, putting 6 — ^ n — 6' m this equation, we obtain {do sm 6 n — 2 { dd Jcos"<9 (n — i)cos''-' Reduction of \sm"'e cos"" dd, 69. If we put dv = sin"' 6 cos 6 ddy we have cos«-^^sin"'+^/9 n — 2{dd , . "^^"^^Jcos^^ • • • • (4) sin"" 6 cos"" 6 dd = m + I + -^ ^^- fsin'«+^6'cos«-^6>^^; . . . (l) m + I } but, if in the same integral we put dv = cos« 6 sin 6 dd, we have sin"'-^<9cos«+' B j. sin"" 6 cos"" e dd n + I m L fsin— 6'cos«+=^^^^. ... (2) § VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 85 When m and n are both positive, equation (i) is not a formula of reduction, since in the new integral the exponent of sin 6 is increased, while that of cos B is diminished. We therefore substitute in this integral sin'«+^ 6 = sin''' d{\ — cos^ (9), so that the last term of the equation becomes ^~ ^ fsin- e cos«-^ ddS- ^LZJL \ sin- d cos« 6>^6'. m + I ] m + I ] Hence, by this transformation, the original integral is repro- duced, and equation (i) becomes fi + ^ 1 f sin- d COS" ede:^ m + I n — I ( + sin- dco^''-' Odd, m -h I J Dividin by I -^ = , we have " m + I m + I sm- 6 cos'' 6 dd=z m -{■ n !^ fsin-6>cos''-^^^^, ... (3) + m a formula of reduction by which the exponent of cos 6 is diminished two units. 86 METHODS OF INTEGRATION, [Art. 69. By a similar process, from equation (2), or simply by put- ting Q — \n — d' m equation (3), and interchanging m and n^ we obtain f . ^ , ^ ,^ sin'«-^l9cos«+^/9 sm.*" d cos" ddd = J ' m + n + ^~ ^ fsin'«-^ d cos« ^^^, . . . (4^ a formula by which the exponent of sin 6 is diminished two units. 70. When 7t is positive and m negative, equation (i) of the preceding article is itself a formula of reduction, for both exponents are in that case numerically diminished. Putting — mm place of m, the equation becomes fcos« l9 _ cos«-^^ n — I fcQs^-^ r/) , >. J sin"' 6 ~ ~ {m— i)sm"'-^d m — i Jsin'«-^ • • • • i5; Similarly, when m is positive and n negative, equation (2) gives {'^de=^ '^r^ _^^£[giBn-%^. ... (6) Jcos«6' (n- I) cos''-' 6 n-ilcos^'-^d ^^ 7(. When m and ;2 are both negative, putting — m and — n in place of m and ;^, equation (3) Art. 69 becomes J sii dd sin''' ^ cos« 6 (m +n) sin'''- ^ (9 cos''+^ 6 n + I f de^ m-^ n] sin'« d cos«+^ ^ ' in which the exponent of cos B is numerically increased. We § VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 8/ may therefore regard the integral in the second member as the integral to be reduced. Thus, putting n in place of » + 2, we derive f dO I J sin"' 6 cos« d~ {n — i ) sin'« - ^ 6 cos'' - ^ 6 m + n — 2 c dd ZJ f ^^ (7) I Jsin'«6'cos«-^^ ' ^^^ Putting 6 = ^7t — d\ and interchanging ;/^ and «, we have [__d^__ _ I J sin'" ^ COS" ^ ~ (;«— i)sin'''~'6'cos''"'6^ m + n-2 f dd m—\ Jsin'«-^(9cos"^ ^ ^ A 72. The application of the formulas derived in the preced- ing articles to definite integrals will be given in the next sec- tion. When the value of the indefinite integral is required, it should first be ascertained whether the given integral belongs to one of the directly integrable cases mentioned in Arts. 27 and 28. If it does not, the formulas of reduction must be used, and if m and n are integers, we shall finally arrive at a directly integrable form. As an illustration, let us take the integral [sm^ecQ^^Bde. Employing formula (4) Art. 69, by which the exponent of sin % is diminished, we have r • 2 /I X fx jn sin ^ cos" Q I f A sm^ Q cos* Odd = ^ + ^ cos* Odd. 88 METHODS OF INTEGRATION, [Art. 72. The last integral can be reduced by means of formula (2) Art. 6']^ which, when m ~ /\^^ gives f X n jn COS^ ^ sin ^ 3 f ^ n in cos* Qde=^ + - cos^ Bdd\ J 4 4J therefore r . 9 /, 6. n jQ sin ^ cos^ 6 cos^ d sin 6 sin 6 cos 6 . ^ I sm^ e cos* 6* ^l9 = -. H + p + — . J 6 24 10 16 73. Again, let the given integral be fcos^^ J sin«6' * By equation (5), Art. 70, we have [ cos^ddd __ _ cos^/^ _ 5 fcos* d do 3 sin^ 6 ~ 2 sin^ 6 2 J sin 6' ' We cannot apply the same formula to the new integral, since the denominator m— i vanishes ; but putting n—4. and m — — i^ in equation (3) Art. 69, we have cos^dde cos^^ [cos^ede f cos* ede _ cos^^ r J sin 6> ~ 2 "^ J 3 J sin (9 cos«^ 3 Jsin e J sin ddO cos^ I + log tan —6+ cos 6. 3 2 Hence [cos^ Odd cos^ 5 cos^ 6* 5, i ^ S /. I — . n n ■ = ^^-n — - — z log tan - ^ — ^ cos 0. J sm^ 6 2 sm^ (^ 6 2^22 § VI.] EXTENSION OF THE FORMULA. 89 Extension of the Formula, 74. Let Y W dx =z ^, (x), U^ {x)dx= ,,(x)y etc., etc. ; then, if the functions ^^ (x), (j),, {x), .... ^« (x), which may be called the successive integrals of (l){x), are known, and also the successive derivatives oi f{x), we shall have J/ W -!> W dx = fix) ^, (.r) - j/' (.r) ^,(^) ^^ = fix) (^) = h Xo and h being constants, /' {x) = - F" {xo+ h - x), f" {x) =: F'" {Xo + h- x), etc. ; and (l>^{x) = x, ^,X^) Hence x^ ;tr^ (j) (x) = , etc. 1-2' ^'"^ ' I-2-3 [f' {xo^h- x) dx =^F' {xo+ h- x)-x+ F" (xo -^ h- x)- x'^ 2 + F''^' (Xo -^h- x) -- — dx. Now F' {xo + /i — x) dx = — F {xo -{- k — x); hence, applying the limits o and /i, we have F(x. 4- /i)= F{xo)+ F' (Xo) h + F" (xo) ^ + 1 ^ Jo + k — x) x"dx 1-2. • n This formula is Taylor's Theorem, with the remainder expressed in the form of a definite integral. §VL] EXAMPLES. 91 / / / y I. •svci-'^ X dxy •' o y 2. sec~^ji:^:r, J o / F • . i/ 5. Q sin Q^O, •'o V 6. cos mB t/B, •'o J o 9. Lcsec-^jpz/jf, Examples VI. [ /V-^-V^ X sin ~ ^ jc + V{i — X ^ sec-^.v — log [x + |/(.r' — i)]. ;r _ log 2 4 2 2»? ^"^ -I ( - tan-^x . 2 2 ;r^f-^ — 2a:f-^ + 2£-^ i [x"^ sec-^ A- — V{^^ — i)]. V 10. J' S sin - + L/0, —9 cos ( - + 6 j + sin ( -+ 6 j 7r//2 92 METHODS OF INTEGRATION, [Ex. VI. / . X sec" X dXy X tan x + log cos x . Lctan'x^/r = \x (^qc^ x — i) dx L xtsinx + log V II 12 J 13. Lv'sinx ^JC, cos X X . 2 2X Sm X -\- 2 cos X — X COS jc. 4. x%m.~^xdxy -x'*sm-'^x\ I sin'' 6 ^0 = — . Jo 2 J, 2j^ 8 ^ / 15. Lr' tan- ^ ^ ^/jc, x'tan-^.r x^ log (i + ^) I 6 "^ ~6 7t 2 \l 16. x'^mr^xdx, -x'sin-^x-l ^(i — Jt:') . Jo o 9 f^ (sin X — cos x)"!"" _ i 17. f-^COSJC^AT, T= 2 2 — lo \ J 18. I f-'^ *^" ^ COS jc dx, cos /? f^ t^" ^ sin (/? + x) 19. e-"^ sm^ xdx\ = - e-^ (i — cos 2x) dx , — (cos 2x — 2 sin 2x — k). 10 ^ ^^ V -I: f ^ sin 6 ^Jo 32 w 30. ^ COS* ^/0, sin COS (8 COS* + 10 cos' + 15) + 150 "Is _ 94/3 4- i:>7r 48 Jo ~ 96 31 32 33 f^os . . -^-5- ^0, J sin' ' J cos' * fsin" ^ IT (2COS'0 . ^0, 7rSin''0 4 ^5- j^(i + cos0r .6 \-^— ^ ' Jsin0cos*0' cos' _ 3 cos _ 3 log tan ^Q 2 sin^ 02 2 sm sm I 4 cos 8 cos G log tan .4 2 J sm 5 sin" K ^ . -, r— — r + - fS — sin cos 0j 3 cos 3 cos 02^ ■" cos-'0 sin cos* 48 - 15^ 32 1 [7 do' ^ 2 J„ cos* 0' ~ I 1,0 H r + log tan - 3 cos cos § VI.] EXAMPLES, 95 [ d^ I 3 cos 6 , 3 , ^ Q '?7. \— ^-s — , — ^^- — ■ c ' 2 h^ log tan-. ^' J sm sm" 20 ' 4 sm' cos 6 8 sin' 8^2 38. Prove that when n is odd H H + log tan ; J sin cos'' /^ — I n — 2> and when « is even f //0 sec«-^0 , sec'^-'Q , . 1 9 -T— —^ = + H- 4- log tan - . J sm cos n — I n — $ 2 { de I . 5 fsec"© , ^ , , ^ e"1 39- -^-3 4- ♦ ^-1 3- + h sec + log tan - . ^^ Jsin'0cos'o* 2Sin'0cos'0 2 [_ s 2 J 4°- J y(.-,^_,) > ^«^^^--sec0. ^xVi^v' - i) + J log [:v: + V{x' - i)]. 41. J (a' - xy dx, ^ '-^ -^ + ^ sm-^ - . a 5 [ dx \ U . ^TT -^ Z 42. T^—, rTs » • — T COS ^/0 = ^^ a- , (Jt:' ^/a: x^pc^ ~ ^) I tan~^ x '{x' ^ i)" 8(x« + ir"^~~8 • 96 METHODS OF INTEGRATION. [Ex. VI. , f cos's — sin'e ,^ r f/ , • . .\ cos — sine ,"1 46. \t-. \id^ = (i + sin cos 6) 7-r— -— ^d^ , ^ J (sin e + cose)' L J ^(sme + cose)' J' sin e cos sine + cose 47. Derive a formula for the reduction of L%' sec** a: ^j^), Art. 10, we have = cot"' I — cot"^ (— i), Jx I +^ in which, if cot"' i is taken as J ;r, cot-'(— i) must be taken as } n. Thus f"' dx I I I + ;r^ 2 Formulas of Reduction for Definite Integrals, 83. The limits of a definite integral are very often such as to simplify materially the formula of reduction appropriate to it. For example, to reduce j: x"" ^-""dx, we have by the method of parts 102 METHODS OF INTEGRATION, [Art. 83. Now, supposing n positive, the quantity ;tr'' £--^ vanishes when x = Oy and also when ;r = 00 [See Diff. Calc, Art. 107 ; Abridged Ed., Art. 91]. Hence, applying the limits o and 00 , x^'B-'' dx = n I x""-"- e-^'dx. By successive application of this formula we have, when n is an integer, ^n e—r ^^ _ ^ ^^ _ jj 2.1. Jo 84. From equation (i) Art. 66^ supposing m > i, we have ■n IT [ sm^^ d dd = ^^^—^ [ sin-- ddO. Jo ^ Jo If m is an integer, we shall, by successive application of this n IT formula, finally arrive at V dd = - or j' sin ^^^ = i, according as m is even or odd. Hence if m is even, f sin" 6 dO = (^ j)(>« - 3) • ■ • • i . 5, . . (p) Jq m{m — 2) 2 2 ^ ^ and if m is odd, f^ sin- d dO = {m - i){m - 3) > - - • 2 _ (p^ Jo m{7n-2) I / ^ § VII.] FORMULAS OF REDUCTION. IO3 85. From equations (3) and (4) Art. 69, we derive l sin"^ e cos« ede = ^ "" ^ f' sin''' 6 cos"-^edd, Jo m + n io - _ IT JT and f' sin'^ 6 cos'^ ^ ^<9 = ^-^LzJL y sm*"-^ d cos« dd, Jo ^^ + n Jo By successive application of these formulas, we shall have for the final integral one of the four forms \^ dd, Kin^^^, Kos^^^, or [' sin ^ cos ^^^. Jo Jo Jo "O The numerator of the final fraction ( or ) is in each case either 2 or i. In the first case, the value of the final inte- gral is J 7t, and the final denominator is 2 : in the second and third cases, the value of the final integral is i, and the final denominator is 3 : in the fourth case, the value of the final integral is J, and the final d.enominator is 4. Therefore (since the factors in the denominator proceed by intervals of 2), it is readily seen that we may write F sin'« 6 cos« 6 dO = (^^-i)(^- 3) •-> (^- 0(^ :^3)^ ,, . (g) provided that each series of factors is carried to 2 or i, and a is taken equal to unity, except when m and n are both even, in which case a = ^ Tt. 104 METHODS OF INTEGRATION. [Art. 86. Elementary Theorems Relating to Definite Integrals, 86. The following propositions are obvious consequences of equation (i^, Art. T^j. ^f{x)dx=-^f{x)dx (I) (f{x)dx=\f{x)dx+(f{x)dx. . . (2) i a J a J c Again, if we put x ^ a ■\- b — z^wq have I" f{x)dx = - { f(a -V b- z) ds =[ f{a + b-z)dz by (i), or since it is indifferent whether we write ^ or ;i; for the variable in a definite integral, \f{x)dx= \/(a-hb-x)dx .... (3) l{ a=c, we have the particular case ^'/{x)dx=^'/{b-x)dx .... (4) § VII.] DEFINITE INTEGRALS. IO5 87. As an application of formula (4), we have ■K It 1t_ V COS- ede=^ cos'«(^ -e\de= [" sin- ddO . . . . (i) IT V Hence the value of ^ cos'" 8 dO as well as that of ^ sin"' 6 dS is given by formulas {P) and {P'). The values of these integrals are readily found when the limits are any multiples of ^ n. For, by equation (2) of the preceding article, we may sum the values in the several quadrants. But, putting 6 =^ k — h ^', and employing equation (i), we have 'sm-^ede=±\ ' cos-'dd0=±\ sin-^dde, . . (2) in which the sign to be used is determined by that of sin'« 6 or cos'" 6 in the given quadrant. In like manner the value of the integral in formula (Q) is numerically the same in every quadrant, and its sign is the same as that of sin'" ^cos'^ ^in the given quadrant. Change of Independent Variable in a Definite Integral, 88. It is often useful to make such a change of independ- ent variable as will leave unchanged, or simply interchange, the values of the limits. As an illustration, let us take the definite integral f- Jo I + I06 METHODS OF INTEGRATION. [Art. 88 If we put X — — , whence log x = — log j, and dx — ^» y r •o u = . 00 logj f+y + l dy = -u; whence we infer that • 00 u = , o log;r dx = 0. 89. Again, let • 00 u = ^^'- Putting ;ir = — , we have _ r2iog^- '°g-^^r- 1 !(->£ ..r. dy Jo a^^f •" ^ )oa^-\-f hence f log log X y _7t lOg^ ;r* 2a Differentiation of an Integral, 90. The integral f {x) dx is by definition a function of x. whose derivative, with reference to jr, is f{x). Thus, putting U= \f{x)dx, i a dU ,, , g VII.] DIFFERENTIATION OF AN INTEGRAL. lO/ This gives the derivative of an integral with reference to its upper limit. By reversing the limits we have, in like manner, when the lower limit is regarded as variable, 91. Now writing the integral in the form U \ u dx ^ (i) if u is a function of some other quantity, «', independent of x and Uy U \s also a function of a^ and therefore admits of a de- rivative with reference to a. From (i) we have dJJ__ dx~^' whence d dU _ dn, da dx da By the principle of differentiation with respect to independent variables [See Diff. Calc, Art. 401 ; Abridged Ed., Art. 200]. d^dU^d_ dU dx da da dx ' Therefore and by integration d dU _ du ^ dx da da ' dU {du J ^ / X dx + C (2) dU __ (du da ~ ]da 108 METHODS OF INTEGRATION. [Art. 9I. Now, in equation (i), C/ is a function of x and a which, when ;r = «, is equal to zero, independently of the value of a. In other words, it is a constant with reference to a^ when x — a\ therefore -r- — o when x =^ a. If, then, we use ^ as a lower da limit in equation (2), we shall have (7 = 0. Therefore dU da du J , . Substituting for x any value b independent of a^ we have ---\ udx — \ -j-u dx , (4) aai a ] a da which expresses that an integral may be differentiated with reference to a quantity of which the limits are independent^ by differentiating the expression under the integral sign. 92. By means of this theorem, we may derive from an inte- gral whose value is known, the values of certain other inte- grals. Thus, from the first fundamental integral, x»dx = - , (l) we derive, by differentiating with reference to n, _ (;2 + i)x''-^^\ogx —x""-^^ {n + 1/ X" log X dx = / .. , , \2 the result being the same as that which is obtained by the method of parts. 93. The principal application of this method, however, is to definite integrals, when the limits are such as materially to § VII.] DIFFERENTIATION OF AN INTEGRAL. IO9 simplify the value of the original integral. Thus, equation (i) of the preceding article gives X'' dx — — ^— , ;2+ I 1: whence, by successive differentiation, I x^ loff X dx= — 7 ^ , Jo (« + 0' 1-2 '^dx^ -, ^, I x^'ilogx) [ x"(\ogxYdx= (- lY ^'^" "'' Integration under the Integral Sign, 94. Let u be a function of jt and a, and let a and ao be con- stants ; then the integral U=\ r[ udA^da, (i) is a function of x and ^, which vanishes when a — 0.0^ inde- pendently of the value of x^ and when x = a., independently of the value of a. From (i) dU [ , ^ d dU whence -y- 3— a ax eta dU [ , — -=1 \ u ax, da Ja therefore -—-— = «, whence -=- = \u da -^ C, dadx dx J no METHODS OF INTEGRATION-. [Art. 94. Now -%— must vanish when a = a^, since this supposition makes ^independent of x; therefore, if we use ^^^ for a lower limit in the last equation, we must have C = 0; therefore dx — u da^ and since u vanishes when x = a, U — \ \ u da ia VJ OL^ Comparing the values of 6^ in equations (i) and (2), we have dx. (2) tc dx da = aja }a}a It is evident that we may also write 2L dx da — ot-Jci ]a]a u da dx. ■ (3) provided that the limits of each integration are independent of the other variable. 96. By means of this formula, a new integral may be de- rived from the value of a given integral, provided we can inte- grate, with reference to the other variable, both the expres- sions under the integral sign and also the value of the inte- gral. Thus, from x"" dx = n + I § VII.] INTEGRATION UNDER THE INTEGRAL SIGN. Ill by multiplying by dn, and integrating between the limits / and 5, we derive whence —^ ax — lop^ . J^ log A' ^7-+ I 96. When the derivative of a proposed integral with refer- ence to rt' is a known integral, we can sometimes derive its value by integrating the latter with reference to vc. Thus, let u — ' — dx (I) In this case da io " «' Jo oc^ hence, integrating, u=— log ^^ + 6^ = log — . . . . (2) since in (i) u vanishes when a—fi. The Definite Integral Regm^ded as the Limiting Value of a Sttm. 97. Let A denote the greatest, and B the least value as- sumed by/(jt'), while x varies from a to b. Then it is evident that t f{x)dx< f Adx; (i) J a J a for, while x passes from a to b, the rate of the former integral 112 METHODS OF INTEGRATION. [Art. 97. is generally less, and never greater than the rate of the latter. In like manner fb fb f{x)dx> \ Bdx (2) J a J a The values of the integrals in the second members of equations (i) and (2) are A {b — a) and B {b — a) respectively. There- fore, if we assume (f{x)dx = M(b-d), (3) we shall have A > M> B. The quantity M in equation (3) is called the mean value of the function /(jr) for the interval between a and b. 98. Let b— a = n Ax\ (4) then the n + i values of x, a, a + AXy a-i-2Ax,--'' b, define n equal intervals into which the whole interval b — a is separated. Let x^^ x^, x„hG n values of x^ one com- prised in each of these intervals; also let 2^/{Xr) Ax denote the sum of the n terms formed by giving to r the n values I • 2 • • • • n in the typical term/(;ir^) Ax; that is, let 2i/{xr) Ax=/{x,yAx -i-/{x^) AX""-{-/{x„) AX, . . (5) § VII.] AN INTEGRAL THE LIMIT OF A SUM. II3 We shall now show that when n is indefinitely increased the limiting value of ^f/(^r) A;r is fix) dx. 99. If we separate the integral into parts corresponding to the terms above mentioned ; thus, Jb ta + AJtr /*« + 2 A J* /{x) dx = f{x) dx + f{x)dx • . . . + f /{x)dx, and let J/j, M^, • • • • Mn denote the mean values of f (x) in the several intervals, we have, in accordance with equation (3), Art. 97, I f{x) dx = Af^Ax -^M^ AX -h M„ AX (6) J a Now, since /(-tv) and Mr are both intermediate in value between the greatest and the least values of /{x) in the inter- val to which they belong, their difference is less than the dif- ference between these values of /(;t'). Therefore, if we put /{Xr) = Mr + er, (7) er is a quantity whose limit is zero when n, the number of intervals, is indefinitely increased, and A;r in consequence diminished indefinitely. Comparing the terms in equations (5) and (6) we have, by means of equation (7), 2l/{x) AX = f{x) dx + (e^-{- e^ + en) t.x. ... (8) 114 METHODS OF INTEGRATION. [Art. 99. Denote by e the arithmetical mean of the n quantities ^1, ^2» • • • • ^« ; that is, let «£ = ^1 + ^2 + ^3 ^« ; (9) then, since e is an intermediate value between the greatest and the least value of ^^, it is also a quantity whose limit is zero when n is indefinitely increased. By equations (9) and (4), equation (8) becomes b [^ 2^ f{x,) t^x = Y^ f{x) dx + e{b- a\ whence it follows that f(x) dx is the limit of ^^/ (^v) dx J a w^hen n is indefinitely increased, since the limit of c is zero. 100. It was shown in the Differential Calculus, Art. 390 [Abridged Ed., Art. 193], that, in an expression for the ratio of finite differences, we may pass to the limit which the ex- pression approaches, when the differences are diminished with- out limit, by substituting the symbol d for the symbol A. The theorem proved in the preceding articles shows that, in like manner, in the summation of an expression involving finite differences, we may pass to the limit approached when the differences are indefinitely diminished, by changing the symbols ^ and A into and d. The term integral, and the use of the long s, the initial of the word sum, as the sign of integration, have their origin in this connection between the processes of integration and sum- mation. VII.] ADDITIONAL FORMULAS OF INTEGRATION. II5 Additional Formulas of Integration. I0(. The formulas recapitulated below are useful in evalu- ating other integrals. {A) and {A') are demonstrated in Art. 17; {B) and {C) in Art. 29; {D) and {E) in Art. 30; {F) in Art. 31 ; (6^) and {G') in Art. 35 ; (//) and (/) in Art. 50 ; {7) in Art. 51 ; {K) in Art. 52 ; {L) in Art. 53 ; {M) in Art. 55 ; (N) and ((9) in Art. 58; {P) and (P') in Art. 84; and (0 in Art. 85. b) a-b log X — a dx f cos^). .,..,..... ((fl J sin 6^ c( cos u logtan^. . ^ ,,:.(/;) C dd , ^ . ^ , I — cos ^ - — 7, = Iog: tan lu — log r — 7^ — ^ == log tan - + - J cos (9 ^ L4 2 L + bcosd Via^~^) r tan log I + sin ^ cos 6 a-b J a ^ b tan ,.]. . . Il6 METHODS OF INTEGRATION. [Art. lOI. [ dB __ I \/{b^a) ^ V{b -d)t^n\d — -— ^ — -\og-^ '- . ....... {H] - log ^~ \ o (/) X V{d^ — x^) a dx '. (?) \^s/(x^ ± d')dx= ^^^'^^ ""'^ ± l' log [x + V{^±a')-\ . . (L) ' (a^ — x^) dx = — sm-^- -\ — {M) dx dx 2sin"^ /^^ (O) Jo Jo mint — 2) 2 2 ^ ^ § VII.] ADDITIONAL FORMULAS OF INTEGRATION. II7 Jo Jo m(ni-2) I ^ ^ (m + n){m + n — 2) ^"^^ in which a= i, unless m and n are both even, when a = ~, 2 Examples VII. I- — T"! ;> [^ > ^> ^^^ ^ ^" integer] -— — ^^ r^. p"^^± - do 2n7t±\7t ^* Jo 2 + COS 9' V3 IT 3. r sin" ear©, 4. sin* fl?i9, 5. J „cos'0^e, 5^ 32" 16 15' 6. I sin^ cos® 6 //G, 512 Il8 METHODS OF INTEGRATION, [Ex. VIL 7. I sin' cos' ^d% ^ ' Jo 4/(1-^')' '" J. 35 sin'« Q dQ , 8. ^ sin"^ e cos"' G ^9, — I si Jo 2'«Jo f x'^" dx ^ 1*3*5 • • ■ (2^ — 1) Ti" Jo 4/(1 — ^"7* 2-4-6 • • • • 2« 2 2'4'6- ■ ' ' 2n 4/(1--^')' 3.5.7. . . (2«+l)* 2tf^ ^^' ' 63- f {x'~a')'^ax , 3;r 1 ^ • I R » 7~ I0« r x' dx 8 '^- JoU^ + ;.')r 15. Prove that o o and derive a formula of reduction for this integral, supposing « > and m '> 1. o n Jo § VII.] EXAMPLES, 119 16. Deduce from the result of Ex. 15 the value of the integral when m is an integer. Jo n\n-\-i) ' ' '[n + m—i) 17. Wa+xY(a-xy ^x. See£x.i6. 2^V2>^ J -a It 18. Tsin' Q (cos 6)^ do. Fut sin'^ G — x, and see Ex. 16. 4504s 5-7. II. 19 19. Show by a change of independent variable that r x^ dx _ r a' dx Jo (a" +x^Y -Jo {a' +xy ' ^ ^ , r x' dx I r dx n and therefore -7-^— — ^vi — ~ t—. — i = — • Jo (« + x^y 2 jo a + x"" 4a (""xjogx^dx log dJ Jo V^ +^ ) 2df r tan-^v. dx 7t^ ^^' ]^x' ^x-\- 1' ' ' 6^3' r . X xdx 7t* 22. tan"^ — 4,4 , -^-7- Jo a x^ -\- a*' i6a' 23. Derive a series of integrals by successive differentiation of the definite integral | f"^ dx. r . 1-2' '-n X^ E-*^ dx = ; . Jo ««-^ ' 120 METHODS OF INTEGRATION, [Ex. VII. m 24. Derive from the result of Art. d"^ ^^ definite integrals (°° „ f > B - '"■* sin nx dx = —. 5 , and f- '«^ cos nx dx = o m -{■ n ' Jo and thence deri . e by differentiation the integrals ^xe- "-sin nxdx ^ -^-.—-^^^ and J^ xe- --cos nx dx = ^^._^^.^ 25. From the results of Ex. 24 derive i:- .«,,-• ^ 2n{sm — n) 2 1 „2\l» » {m' + 7i') 1: „ , 2m (m — 3n) x^ £- '"^ cos nx dx = — 7-H ^Ts" • (m' + n) 26. From the fundamental formula (k') derive (dx _ TT and thence derive a series of formulas by differentiation with refer- ence to a. dx 71 1-3 •• • (2« — 3) I 27. Derive a series of integrals by differentiating with reference to /5, the integral used in Ex. 26. p x^'^-^dx _ 7t i'3-5 > • (2;^ — 3) I Jo (a + l^x'V ~ 2«a:i i-2-3~- • • (« - i) 'yS«-* * § VIIJ EXAMPLES. 121 28. From the integral employed in examples 26 and 27, derive the value of -. ; — Tr-arr • Jo (n; + ^^ ) ^ Differentiate tivice with reference to /?, ««^ ^«^^ a///^ reference to a. f x* dx __ i-3'i re 29. Derive an integral by differentiation, from the result of Ex, II., 67 Jo (^-^ + ^0 (^' + ay ~ ^a'b {a + bf ' 30. Derive an integral by integrating —, ^ = — . J o a ~T' X 2a fTtan-.^-tan-.^l^ = ^log^. JoL ^ xj X 2 ^ g 31. Derive a definite integral by integrating 1:- sin nx ax = —5 5 /« + n with reference to n. m^ +^' (cos «:\: — cos bx) ax — — \oa^ ]o X 2 m 32. Derive a definite integral from the integral employed in Ex. 3: by integration with reference to m. 22 METHODS OF INTEGRATION. [Ex. VIL T^2,' Derive an integral by integrating with respect to m ffi €- ^"^^ COS nx dx = —T, — ^ COS nx ax^= — log -^ 34. Derive an integral by integrating with respect to n the integral used in the preceding example. re-'"^ . . . . X ^ m(a- b) ■ (sin ax — sin bx) dx = tan" ' — ^ / Jo ^^ ^ m' + ab m^ + ab ' 35. Show by means of the result of Ex. 32 that (•00 • sm nx . TT ax = — X 2 $6. Derive an integral by integration from the result of Ex. II., 67. (CO 2 1 ' 2 log 2 ^dx by the method of Art. 96. 7t(a — b). 38. Evaluate log i + — , logxdx. it a (log^ ~ i). §V1II.] PLANE AREAS. I23 CHAPTER III. Geometrical Applications. VIII. Plane Areas, 102. The first step in making an application of the Inte- gral Calculus is to express the required magnitude in the form of an integral. In the geometrical applications, the magni- tude is regarded as generated while some selected independ- ent variable undergoes a given change of value. The inde- pendent variable is usually a straight line or an angle, varying between known limits ; the required magnitude is either a line regarded as generated by the motion of a point, an area generated by the motion of a line, or a solid generated by the motion of an area. A plane area may be generated by the motion of a straight line, generally of variable length, the method selected depending upon the mode in which the boundaries of the area are defined. An Area Generated by a Variable Line having a Fixed Direction, 103. The differential of the area generated by the ordinate of a curve, whose equation is given in rectangular coordinates, has been derived in Art. 3. The same method may be em- ployed in the case of any area generated by a straight line whose direction is invariable. 124 GEOMETRICAL APPLICATIONS. [Art. I03. Let AB be the generating line, and let R be its intersection with a fixed line CD^ to which it is always perpendicular. Suppose R to move uni- formly along CDy and let RS be the space described by R in the interval of time, dt. U D Then the value of the differential of the ! area, at the instant when the generating line passes the position AB^ is the area which would be generated in the time dt^ if the rate of the area were constant. This rate would evidently become constant if the generating line were made constant in length ; and therefore the differential is the rectangle, represented in the figure, whose base and altitude are AB and RS ; that is, it is the product of the generating line^ and the differential of its motion in a direction perpendicular to its length. 104. In the algebraic expression of this principle, the inde- pendent variable is the distance of R from some fixed origin upon CD, and the length of AB is to be expressed in terms of this independent variable. When the curve or curves defining the length of AB are given in rectangular coordinates, CD is generally one of the axes; thus, if the generating line is the ordinate of a curve, the differential is y dx, as shown in Art. 3. It is often, how- ever, convenient to regard the area as generated by some other line. For example, given the curve known as the witch, whose equation is ^ X — 2af' -\- ^X r=. o (i) This curve passes through the origin, is symmetrical to the axis of X, and has the line x = 2a for an asymptote, since X = 2a makes y = ± 00 . Let the area between the curve and its asymptote be re- \ § VIII.] AREAS GENERATED BY VARIABLE LINES. 125 quired. We may regard this area as generated by the line PQ parallel to the axis of ;r, y being taken as the independent variable. Now PQ = 2a — Xy hence the required area is A = \ {2a- x)dy . . . . (2) From the equation (i) of the curve, we have _ 2a^ whence 2a — x M Fig. 4. and equation (2) becomes ^ = 8^r ^^^ - = 4/y^tan-^J^T =4;r^^ J_«,/+4^?2 ^ 2tf L« Oblique Coordinates. (05. When the coordinate axes are oblique, if a denotes the angle between them, and the ordinate is the generating line, the differential of its motion in a direction perpendicular to its length is evidently sin a-dx ; therefore, the expression for the area is ^ = sin «f M/ dx. 126 GEOMETRICAL APPLICATIONS. [Art. I05. As an illustration let the area between a parabola and a chord passing through the focus be required. It is shown in treatises on conic sections, the expression for a focal chord is AB — \a(iosQ(?a ^ . . . (i) X where a is the inclination of the chord to the axis of the curve, and a is the distance from the focus to the vertex. It is also shown that the equation of the curve referred to the diameter which bisects the chord, and the tan- gent at its extremity which is parallel to the chord is j^ — 4^ cosec^ a-x (2) The required area may be generated by the double ordi- nate in this equation; and since from (i) the final value of J/ is ± 2^ cosec^ oc, equation (2) gives for the final value of x OR = a cosec^ a. Hence we have Fig. 5. (a cosec^a y4 = 2 sin «f y dx^ J o or by equation (2) (a cosec^a \/xdx = o Sa^ cosec^ a 3 Employment of an Auxiliary Variable, 106. We have hitherto assumed that, in the expression A ydx, §VIII.] EMPLOYMENT OF AN AUXILIARY VARIABLE. 12J X is taken as the independent variable, so that dx may be assumed constant ; and it is usual to take the limits in such a manner that dx is positive. The resulting value of A will then have the sign of j, and will change sign if y changes sign. It is frequently desirable, however, as in the illustration given below, to express both y and dx in terms of some other variable. When this is done, it is to be noticed that it is not necessary that dx should retain the same sign throughout the entire integral. The limits may often be so taken that the ex- tremityof the generating ordinate must pass completely around a closed curve, and in that case it is easily seen that the com- plete integral, which represents the algebraic sum of the areas generated positively and negatively, will be the whole area of the closed curve. 107. As an illustration, let the whole area of the closed curve f I, ©' * (f) represented in Fig. 6, be required. If in this equation we put we shall have ©'= cos tp ; whence ^ = ^ sin^ //', and y = b cos^ ip , . . (i Therefore \y dx = ^ab cos^ ip sin^ ^ dip. 128 GEOMETRICAL APPLICATIONS, [Art. I07. Now if in this integral we use the Hmits o and 27r, the point determined by equation (i) de- scribes the whole curve in the direction A BCD A. Hence we have for the whole area (277 cos^ ^ sin^ ^ di\)^ and by formula (0 3-I-I _ '^^nab ^ 6-4-2 8 The areas in this case are all generated with the positive sign, since when j/ is negative dx is also negative. Had the generating point moved about the curve in the opposite direc- tion, the result would have been negative. Area generated by a Rotating Line or Radius Vector. (08. The radius vector of a curve given in polar coordinates is a variable line rotating about a fixed extremity. The angular rate is denoted by ^ and may dt be re- garded as constant, although the rate at which area is generated by the radius vector OPy Fig. 7, is not constant, be- cause the length of OP is not constant. The differential of this area is the area which would be generated in the time dt, if the rate of the area were con- stant ; that is to say, if the Fig. 7. radius vector were of constant VIII.] AREAS GENERATED BY ROTATING LINES. 1 29 length. It is therefore the circular sector OPR of which the radius is r and the angle at the centre is dd. Since arc PR = r dd, sector OPR^-r" dd\ 2 ' therefore the expression for the generated area is (I) 109. As an illustration, let us find the area of the right-hand loop of the lemniscata 7^= a^ cos 26. Fig. 8. The limits to be employed are those values of 6 which make r = o ; that is and -. 4 4 Hence the area of the loop is -=?/:= 9 COS 20 dd = - sin 26 4 110. When the radii vectores, r^ and r^ corresponding to the same value of 6 in two curves, have the same sign, the area generated by their difference is the difference of the polar areas generated by r^ and r^. Hence the expression for this area is \ =ii<'.' ri') dd. (2) I30 GEOMETRICAL APPLICATIONS. [Art. III. 111. Let us apply this formula to find the whole area between the cissoid Tx =■ 2a (sec B — cos 6), Fig. 9, and its asymptote BP2y whose polar equation is ^2 = 2a sec 0. One half of the required area is generated by the line PtP2, while 6 varies from o to I TT. Hence by the formula Fig. 9. A = 2^2 J^' (2-cos2^) d6 = ^7ra\ Therefore the whole area required is ^Tta^. Transformation of the Polar Formulas, 112. In the case of curves given in rectangular coordinates, it is sometimes convenient to regard an area as generated by a radius vector, and to use the transformations deduced below in place of the polar formulas. Put y = fnx ; (I) now taking the origin as pole and the initial line as the axis of ,r, we have X = r cos 6, therefore and y — r ^\ViQ\ . . • (2) ==^=tan^, X dm = sec^ dd (3) m § VI 1 1.] TRANSFORMA TION OF THE POLAR FORMULA S. 1 3 1 From equations (2) and (3), j(^ dm = r^ dO ; therefore equation (i) of Art. 108 gives A = — Li'^ dm. ...... (4) In like manner, equation (2) Art. 1 10 becomes A^\\{xi-x?)dm (5) (13. As an illustration, let us take the folium ^ ;r^ + y — 3«;rj/ = O (i) Putting y = mx^ we have :t^ ( I -\- if^) — lamx^ —o (2) This equation gives three roots or values of ;r, of which two are always equal zero, and the third is ;r = J^; <4) whence _ ^am ,. These are therefore the coordinates of the point P in Fig. 10. o Since the values of x and y vanish when m = o, and when / m = c^ , the curve has a loop in the first quadrant. To find \ 132 GEO ME TRIG A L A PPLIGA TIONS. [Art. 113. the area of this loop we therefore have, by equation (4) of the preceding article, 2 n^ dm Jo(i + (i + n^f 3^ 2 I + m^A^ 2 114. The area included between this curve and its asymp- p tote may be found by means of equation (5), Art. 112. The equation of a straight line is of the form D\ c y = mx + by Fig. 10. and since this line is parallel to ^ = mx^ the value of m for the asymptote must be that which makes x and y in equations (4) and (5) infinite ; that is, //^ = — I ; hence the equation of the asymptote is y Ar X — b, (6) in which b is to be determined. Since when m= — i, the point P of the curve approaches indefinitely near to the asymp- tote, equation (6) must be satisfied by P when m= — i. From (4) and (5) we derive m^ 4- m xain y -\- X = ^a — ■ -„ = ^ r ; whence, putting m = — i, and substituting in equation (6) — a = by the equation of the asymptote AB, Fig. 10, is y -\- x= -a c (7) § VI 1 1 .] TRANSFORMA TION OF THE POLAR FORMULA S* 1 3 3 Now, as m varies from — oo to o, the difference between the radii vectores of the asymptote and curve will generate the areas OBC and ODA, hence the sum of these areas is repre- sented by A = -\ {xi — x^) dm, lly-- in which Xc^ is taken from the equation of the asymptote (7), and Xy_ from that of the curve. Putting J = mx, in (7), we have a X2= - I + m and the value of x^ is given in equation (4). Hence ^ ^ r 3 L_T 1 + ;;r m 2 I ~ 7n + m G o 4f ^. Adding the triangle OCD, whose area is \a^, we have for the whole area required ^d^. * This reduction is given to show that the integral is not infinite for the value m= — I, which is between the limits. See Art. 77. 134 GEOMETRICAL APPLICATIONS. [Ex. VIII. Examples VIII. I. Find the area included between the curve /d the axis of x, 2. Find the whole area of the curve a^y^ ■- x^ (a- — ^'). \J 3. Find the area of a loop of the curve 13 J 4. Find the area between the axes and the curve y{x' ■ha'')=d'{a-x). ^^ p - ^^1 5. Find the area between the curve 22. 22 22 xy + ay — a x^ = o, and one of its asymptotes. ' 2«*. ^ / 6. Find the area between the parabola y = ^ax and the straight line y = X. ^, — , 3 7. Find the area of the ellipse whose equation is ax^ + 2bxy + cy' = i. ^,(J- i'y / §VIIL] EXAMPLES. 135 i-^ .. . — _ _ \y 8. Find the area of the loop of the curve cy" = (x — d){x — by J in which ^ > o and b y a. 8 (^ - a )% \/ 9. Find the area of the loop of the curve ay = X* (b + x). 10. Find the area included between the axes and the curve 105^8 =i / V \ ^ ab I. — (:-)"-©• 20 \. II. If « is an integer, prove that the area included between the axes and the curve &-& = . n(n— 1) ' ' ' 1 , IS A = — r-^ — - — ; — c ab. 12. If n'ls an odd integer, prove that the area included between the axes and the curve \n(n— 2) '♦'!]' nab IS A — . . 271 \2n — 2) • • • 2 2 136 GEOMETRICAL APPLICATIONS. [Ex. VIII. 13. In the case of the curtate cycloid X = aip — b ?>m tj.^ y = a — b cos ^, find the area between the axis of x and the arc below this axis. (2a' + b") cos-^l - za Vib' - «"). 14. li b— iaTTj show that the area of the loop of the curtate cycloid is 15. Find the area of the segment of the hyperbola ^j^-^ > X = a sec ip, y = b tan t/j^ cut off by the double ordinate whose length is 2b. ab V2 16. Find the whole area of the curve r"" — a" cos' ■¥ b"" sin" 9. 17. Find the area of a loop of the curve r" = d' cos' — ^' sin'' 0. log tan ^] 2 ^ ab (a'-b') _,a — + ^^ ^ tan - 22 b \^ 18. Find the areas of the large and of each of the small loops of the curve r — a cos Q cos 2O : § VI 1 1.] EXAMPLES, 137 and show that the sum of the loops may be expressed by a single integral. s/. nc^ . a . 7ta a -J- + - , and . 16 4 ' 32 8 12 9. In the case of the spiral of Archimedes, ^ 3 ^ ^ find the area generated by the radius vector of the first whorl and that generated by the difference between the radii vectores of the «th and (n + i)th whorl. ^, and 8««V.-''^'« 6 20. Find the area of a loop of the curve r = a sm 39. 21. Find the area of the cardioid r = 4a sin' ^. 6;r«'. 22. Find the area of the loop of the curve cos 29 a^ (4 — Tt) r = a^^ — -. -. cos 9 2 23. In the case of the hyperbolic spiral, rB = ay show that the area generated by the radius vector is proportional to the difference between its initial and its final value. 138 GEOMETRICAL APPLICATIONS. [Ex. VI I L 24. Find the area of a loop of the curve r^=^ a cos n 9. . 25. Find the area of a loop of the curve „ _ . sin 38 • d" 7 — u a fl • — • COS o 2 26. Find the area of a loop of the curve r' sin = <3!* cos 28. Notice that r z> ?ra/ and finite from = ^ to^ = — , and that — — •^ •' 4 4 ' J sm Q is negative in this interval. d\ ^2 ~ log (i + ^2) . » / 27. Find the area of a loop of the curve X (x'^yy^a^xy. Transform to polar coordinates. — . 28. In the case of the lima9on r = 2a cos B + If, find the whole area of the curve when b> 2a and show that the same expression gives the sum of the loops when /' < 2a. (2a^ + ^');r. § VIII.] EXAMPLES, 139 29. Find separately the areas of the large and small loops of the lima9on when b < 2a. If o' = cos-M — — ) , large loop = (2^' + h') a -V ^ V(4«' - b'') ; small loop = {^2d' + b"") {n — a) — ^ ^(4^^ - b'\ 30. Find the area of a loop of the curve /-' r= d^ cos n^ + /$^ sin « 9. 31. Find the area of the loop of the curve 2 cos 2 6' — I \/{a' + b') r = a cos [5V3-|->. 32. Show that the sectorial area between the axis of x^ the equi- lateral hyperbola -r' -/ = I, and the radius vector making the angle 6 at the centre is represented by the formula . I - I + tan ^ = - log ; 4 ^ I — tan Q ' and hence show that f2A _|_ ^ - 2A g2A f-2A .V = , and y = . 2 2 If A denotes the corresponding area in the case of the circle x' +/ = I, we have X = cos 2^, and y = sin 2^. HO GEOMETRICAL APPLICATIONS. [Ex. VII L In accordance with the analogy thus presented^ the values of x and y given above are called the hyperbolic cosine and the hyperbolic sine of 2 A. Thus f2A ^_ ^-2A ^_2A f2A — := cosh (2^), — sinh (2A). \^ 33. Find the area of the loop of the curve JI-* ^" 3«^y + 2«y = o. ^^ 34. Find the area of the oop .of the curve . -^ s'^' \ 38. Trace the curve • y AT = 2^ sin — , X 35" lyfj -f- T ;»;2« + i4.jj,2« + i — (2«+ i) «Jt:«_y«. - — a^ , 35. Find the area between the curve jp2« + i _|_^2«+i — (^2n + i) axy*" . . 2« + I „ and its asymptote. a . 36. Find the area of the loop of the curve y^ + ax^ — axy = o. 37. Find the area of a loop of the curve x^ 4- jj;* = c^xy. 60 and find the area of one loop. §IX.] VOLUMES OF GEOMETRIC SOLIDS. 141 IX. J^'^~ Volumes of Geometric Solids. x^ 115. A geometric solid whose volume is required is fre- quently defined in such a way that the area of the plane sec- tion parallel to a fixed plane may be expressed in terms of the perpendicular distance of the section from the fixed plane. When this is the case, the solid is to be regarded as generated by the motion of the plane section, and its differential, when thus considered, is readily expressed. 116. For example, let us consider the solid whose surface is formed by the revolution of the curve APB^ Fig. ii, about the axis OX. The plane section per- pendicular to the axis OX \s a circle; and if APB be referred to rectangu- lar coordinates, the distance of the section from a parallel plane passing- through the origin is jr, while the radius of the circle \s y. Supposing the centre of the section to move uniformly along the axis, the rate at which the volume is generated is not uniform, but its differential is the vol- ume which would be generated While the centre is describing the distance dx, if the rate were made constant. This differen- tial volume is therefore the cylinder whose altitude is dx, and the radius of whose base isj^. Hence, if F denote the volume, dV = Tty^ dx. 117. As an illustration, let it be required to find the volume of the paraboloid, whose height is h^ and the radius of whose base is b. 142 GEOMETRICAL APPLICATIONS. [Art. 1 1 7. The revolving curve is in this case a parabola, v/hose equa- tion is of the form and since y = b when x — h, U^ — ^ky whence 4<^ =r -7 ; ft the equation of the parabola is therefore , ^ f = -^.. Hence the volume required is y— TV \ y"^ dx = n —\ x dx nb'h 2 1(8. It can obviously be shown, by the method used in Art. 116, that whatever be the shape of the section parallel to a fixed plane, the differential of the volume is the product of the area of the generating section and the differential of its ^notion perpendicular to its plane. If the volume is completely enclosed by a surface whose equation is given in the rectangular coordinates x, y, 2, and if we denote the areas of the sections perpendicular to the axes by Ajr, Ay, and A^., we may employ either of the formulas V = \a^. dx, V ^ [a,, dy, V=:\a, dz. The equation of the section perpendicular to the axis of x is determined by regarding x as constant in the equation of the surface, and its area A^ is of course a function of x. § IX.] VOLUMES OF GEOMETRIC SOLIDS. 1 43 For example, the equation of the surface of an ellipsoid is The section perpendicular to the axis of x is the ellipse f ^ __c? — x^ b c whose semi-axes are- ^(c^ — x^) and - V{a^ — ^. Since the area of an ellipse is the product of tt and its semi- axes, The limits for x are ±a, the values between which x must lie to make the ellipse possible. Hence nbc f'' , 2 .Ji\ ^.. _^7tabc x^) dx 119. The area A.r can frequently be determined by the con- ditions of the problem without finding the equation of tbe surface. For example, let it be required to find the volume of the solid generated by so moving an ellipse with constant major axis, that its center shall describe the major axis of a fixed ellipse, to whose plane it is perpendicular, while the ex- tremities of its minor axis describe the fixed ellipse. Let the equation of the fixed ellipse be 144 GEOMETRICAL APPLICATIONS. [Art. 119. and let c be the major semi-axis of the moving eUipse. The minor semi-axis of this ellipse is y. Since the area of an ellipse is equal to it multiplied by the product of its semi-axes, we have Ax= 71 cy — — ^/ic^ — x^), a Ttbc r^ Therefore V= — V{c^ — j^)dx\ hence, see formula (^), F = '^abc The Solid of Revolution regarded as Generated by a Cylindrical Surface, 120. A solid of revolution may be generated in another manner, which is sometimes more convenient than the employment of a circular section, as in Art. 116. For example, let the cissoid PORy Fig. 12, whose equation is 1 1 \ J^. l^ — ^ A ■ — ~c p v^__ \ V Fig. 12. pass from the value OA will evidently generate the solid of revolution y^ {2a — x) = A^f revolve about its asymptote AB. The Hne PR, parallel to AB and terminated by the curve, describes a cylindrical surface. If we con- ceive the radius of this cylinder to 2a to zero, the cylindrical surface Now every §IX.] DOUBLE nVTEGRATIOJV. 145 point of this cylindrical surface moves with a rate equal to that of the radius; therefore the differential of the solid is the product of the cylindrical surface, and the differential of the radius. The radius and altitude in this case are PC=2a therefore Putting x, and (•za 2ax fxdx. .^\i X — a = a sin 6, PR=2y, u V = 47ra^ ' (cos^ + cos^ 6 sin 6) dd = 2;rV. Double Integration, 121. When rectangular coordinates are used, the expression for the area generated by a line parallel to the axis of y and terminated by two curves is A^\^^{y^-y^dx. (I) Let AB, in Fig. 13, be the initial, o a c and CD the final position of the gen- ^^^- ^3- erating line, then the area is ABDC, which is enclosed by the curves and by the straight lines a, x=b. 14^ GEOMETRICAL APPLICATIONS. [Art. 121. If in equation (i) we substitute for y^ —ji the equivalent ex- pression cfyf we have (2) which expresses the area in the form of a double integral. In this double integral the limits ji and j/g forj/, are functions of .r, while a and b^ the limits for x, are constants. 122. If the area is that of a closed curve y^ and y2 are two values of J/ corresponding to the same value of x in the equa- tion of the curve, and a and b are the values of x for which y^ andjj/g become equal, as represented by the dotted lines in Fig. 13. It is evident that the entire area may also be expressed in the form A-^\^^yxdy; (3) and that when either of the forms (2) or (3) is applied to the area of a closed curve the limits are completely determined by the equation of the curve. 123. The limits in either of the expressions (i) or (2) define a certain closed boundary, and since either of these integrals represents the included area, it is evident that we may write II dy dx dxdy\ provided it is understood that the limits in the two expressions are such as to represent the same boundary. It should however be noticed that if the boundary is like that represented by the full lines in Fig. 13, or if the arcs j/=j/i and y =/2 do not belong to the same curve, we cannot make a practical application of the form (3) without breaking up t'he integral into several parts. §IX.] DOUBLE INTEGRATION. 147 124. Let ^ (-i',j) be any function of x and j. In the double integral f f ' {x,y)dy dx, (i) J a J J2 X is considered as a constant or independent of y in the first integration, but the limits of this integration are functions of x. The double integration is then said to exte7id over the area which is represented by the expression f ^'' dyd,x or J^ (72-^1) dx. (2) 125. Now let the surface, of which z = ^ (x, y) (3) is the equation in rectangular coordinates, be constructed ; and let a cylindrical surface be formed by moving a line perpen- dicular to the plane of xy about the boundary of the area (2) over which the integration extends. Let us suppose the value of z to be positive for all values of x and y which represent points within this boundary. Then the cylindrical surface, together with the plane of xy and the surface (3), encloses a solid, of which the base is the area (2) in the plane xy, or ASBR in Fig. 14, and the upper surface is CQDP a 2)ortion of the surface (3). Let SRPQ be a section of this solid perpendicular to the axis of x. In this section x has a constant value, and the ordinates of R and 5 are the corresponding values of y^ and ja- The area of this section, which denote Fig. 14. 148 GEOMETRICAL APPLICATIONS. [Art. 1 25. by A^^ as in Art. 1 1 7, may be regarded as generated by the line z, hence A and therefore iy\ (b rja \ zdydx (i) which is identical with expression (i) Art. 124. 126. Now it is evident that the same volume may be ex- pressed by = \zdxdy^ provided that the double integration extends over the same area. Hence, with this understanding, we may write # (jt, j) dy dx = (f) {x, y) dx dy. In this formula x and y may be regarded as taking the places of any two variables, the limits of integration being determined by a given relation between the variables. Thus we may write (j) {u, v) dv du = \\(}) {u, v) du dv, provided the limits of integration are determined in each case by the same relation between u and v. 127. For example, if this relation is U^ ->r V^ — c"^ =. Qj § IX.] DOUBLE INTEGRATION. 149 the range of values in the first integration is between that is, we must have or 1^ ■¥ v^ — c^ o. From equation (3) we find the limits for j/ to be hence, by formula {M), equation (2) becomes V = n {c^ — j^) dx. Finally the limits for x are ± a^ since y is real only when x is between these limits ; therefore V c^x x^ I — - 7ta^ , Elements of Area and Volume. 131. In accordance with Art. 100, the expression for an area, J J dydx, (i) IS the limit of the sum 2li2',\i^y-\c.x. Since each of the terms included in ^^"^ Ay is multiplied by the common factor ax, this sum may be written in the form Sl^^-jA/A^- (2) § IX.] ELEMENTS OF AREA AND VOLUME. 1 53 The sum (2) consists of terms of the form A7 A;r ; and this product is called the element of the sum ; in like man- ner, the product dy dx, which takes the place of t\y Ax when we pass to the limit by substituting integration for summation, is called the element of the integral (i), or of the area represented by it. 132. We may now regard the process of double integration as a process of double summation, as indicated by expression (2), followed by the act of passing to the limiting value. In the first summation indicated, the elemental rectangles corre- sponding to the same value of x are combined into the term (^2 ~ y'x) ^-^'j which may be called a linear element of area, since its length is independent of the symbol A. 133. It is easy to see that, in a similar manner, when rec- tangular coordinates are used, a volume may be regarded as the limiting value of the sum of terms of the form A,t' Ay Az\ and hence dx dy dz^ which takes its place when we pass to the limiting value by substituting integration for summation, is called the element of volume. If the summation is effected in the order ^, 7, x, the first operation combines the elements which have common values of y and x into the linear element of volume, (Z2- zi) AX Ay, 154 GEOMETRICAL APPLICATIONS. [Art. 1 33. The second operation combines the linear elements correspond- ing to a common value of x^ over a certain range of values oi y, into a term whose limiting value takes the form A^ AX. This last expression represents a lamina perpendicular to the axis of X, whose area is A^, a section of the solid, and whose thickness is A.r. Polar Elements, 134. If in the formula for a polar area, \rl-r^)de, (I) A = '- 2 J [equation (2), Art. no], we substitute for -{r^— rf) the equiv- r dr, we obtain A=\^ [\drde, (2) in which a and ^ are fixed limits for 6. Now it follows, from Art. 126, that the limits being deter- mined by a certain relation between r and 6, this integral may also be put in the form A=[ r\''de'dr=\\{e^-e^)dr, ... (3) i a h^ J a § IX.] POLAR ELEMENTS. 1 55 in which a and b are the limiting values of r, between which 6 is possible. The expression r dr dOy in equation (2), is called th.^ polar element of area.* 135. The formula A = \r{e,-e^dr -\r{e,-e,) may also be derived geometrically ; for r (S^ — 6^ is the length of an arc whose radius is r. As r increases, this arc generates the surface, and it is plain that every point has a motion, whose differential is dry in a direction perpendicular to the arc. 136. In determining the volume of a solid, it is sometimes convenient to express ;? as a function of the polar coordinates of its projection in the plane of xy. In this case we employ the linear element of volume, (<8'2 — ^1) r dr ddy corresponding to the polar element of area. * It is easily shown that the area included between the circles whose radii are rand r + Ar, and the radii whose inclinations to the initial line are 6 and fl + A« is (r+ ^Ar) Ar Afl. Since r 4- i A r is intermediate between r and r + A r, the limiting value of the sum, of which this is the element, is, by Art. 99, the integral of the element rdrde. In the summation corresponding to equation (i), the elements are first combined into the sectorial element while in the summation corresponding to equation (3), they are first combined into the arc-shaped element (r+ iArX^a -0,) An 156 GEOMETRICAL APPLICATIONS. [Art. 136. As an illustration, let us determine the volume cut from a sphere by a right cylinder, having a radius of the sphere for one of its diameters. Taking the centre of the sphere as the origin, the diameter of the cylinder as initial line, and the axis of z parallel to the axis of the cylinder, we have for every point on the surface of the sphere (I) where a is the radius of the sphere. Hence •^2 — -S"! = 2 4/(^2 — r^). and V -n> T^Yrdrde-^ L 3 (^2 - 7^)^ dd. The circular base passes through the pole, and its equation is r — a cos 6, (2) hence the limits for r are o and a cos 6, and by substitution we obtain V^—Ui -sin'6)de. The limits for are ± — , the values which make r vanish in equation (2) ; but it is to be noticed that the expression (d^ — r^)t, for which we have substituted <^ sin^ d, is always posi- tive^ whereas sin^ S is negative in the fourth quadrant. Hence the value of V is double the value of the integral in the first quadrant ; that is, V^^\' {i-^m^B) dd 27ta'' 3 8^ 9 §IX.] POLAR ELEMENTS, 157 If a second cylinder whose diameter is the opposite radius of the sphere be constructed, the whole volume removed from the sphere is ^Ttar 1 6^3 , and the portion of the sphere which [6^ remains is , a quantity commensurable with the cube of 9. the diameter. Polar Coordinates in Space, 137. A point in space may be determined by the polar coordinates p, ^^ and ^, of which p de- notes the radius vector OP, Fig. 15, ^ the inclination POR of p to a fixed plane passing through the pole, and d the angle ROA, which the projection of p upon this plane makes with a fixed line in the plane. The angles ^ and d thus correspond to the latitude and longitude of the point P considered as situated upon the surface of a sphere whose radius is p. The radius of the circle of latitude BP is Fig. 15. PC = p cos (j>. The motions of P, when p, ^, and Q independently vary, are in the directions of the radius vector OP and of the tangents at P to the arcs /7? and PB, The differentials of these motions are respectively dp. p dd ; 158 GEOMETRICAL APPLICATIONS. [Art. 1 37, and since these motions are mutually rectangular, the element of volume is their product, f^ cos ^ dp d dQy and F^fffp^cos^^p^^^^ (i) (38a Performing the integration with respect to /o, the for- mula becomes V^-\\{9l- pl)zosii>d(i>de (2) When the radius vector lies entirely within the solid, the lower limit f\ must be taken equal zero, and we may write V=-\\p'zo->^dde (3) The element of this double integral has the form of a pyramid with vertex at the pole. If, on the other hand, in formula (i) we perform first the integration with respect to ^, we have F=|j(sin4-sin#i)pVp^^. .... (4) Taking the lower limit ^1 = o, so that the solid is bounded by the plane OR A, we have the simpler formula V--=Usm(f>fJ'dpdd (5) 139. The formulas of the preceding article take simpler § IX.] POLAR COORDINATES IN SPACE. 159 forms when applied to solids of revolution. Let OZ, Fig. 15, be the axis of revolution, then p and 6 are polar coordinates of the revolving curve, OR being the initial line. Now 6 is in this case independent of p and ^, and its limits are o and 27t. The integration with reference to d may therefore be performed at once. Thus from (3) we obtain V =t) f? COS ^ d(l> ; (6) and in each of the formulas the factor 27r may take the place of the integration with reference to 6. 140, As an example of the use of equation (6), let us find the volume generated by a circle revolving about one of its tangents. The initial line, being perpendicular to the axis of revolution, is a diameter ; hence if a is the radius of the circle its equation is ft — 2a cos ^, and the limits for ^ are and -. 22 Substituting in (6) cos^ (j) dcj) — 27i^a^. 141. The following example of the use of equation (4), Art. 138, is added to illustrate the necessity of drawing a figure in, each case to determine the limits to be employed. Let it be required to find the volume generated by the revolution of the cardioid about its axis, the equation of the curve being p = a{i +sin^), . . Fig. 16. (I) l6o . GEOMETRICAL APPLICATIONS. [Art. I4I. when the initial line is perpendicular to the axis of the curve, as in Fig. 16. The figure shows that the upper limit for ^ is — ;r, while the lower limit is the value of ^ given by equa- tion (i) ; therefore sin (^0 = I, and sin (j^. = i. a The limits for ft are evidently o and 2a. Substituting in equa- tion (4) Art. 138, 21. , 3 4^Jc Examples IX. I. Find the volume of the spheroid produced by the revolution of the ellipse, about the axis of x. . 2. Find the volume of a right cone whose altitude is a, and the radius of whose base is d. nab^ 3. Find the volume of the solid produced by the revolution about the axis of x of the area between this axis, the cissoid y {2a — x) = x\ and the ordinate of the point («, a). S^'tt (log 2 — |). § IX.] EXAMPLES, l6l 4. Find the volume generated by the revolution of the witch, y^x — 2ay^ + ^d^x = o, about its asymptote. See Art. 104. dfTt^a^. 5. The equilateral hyperbola x' — y^ — m revolves about the axis of .v : show that the volume cut off by a plane cutting the axis of x perpendicularly at a distance a from the vertex is equal to a sphere whose radius is a. 6. An anchor ring is formed by the revolution of a circle whose radius is b about a straight line in its plane at a distance a from its centre: find its volume. i>. > 0- 2n'^aB\ 7. Express the volume of a segment of a sphere in terms of the altitude h and the radii a^ and a^ of the bases. — {le + 3^r -I- 3«/). 8. Find the volume generated by the revolution of the cycloid, x = a(ip — 'SAiiip), y = a {i — cos ^■), about its base. 57^''^^ 9. The area included between the cycloid and tangents at the cusp and at the vertex revolves about the latter ; find the volume gen- erated. 10. Find the volume generated by the revolution of the part of the curve y=8-, which is on the left of the origin, about the axis of x. 7t 1 62 GEOMETRICAL APPLICATIONS. [Ex. IX 11. The axes of two equal right circular cylinders, whose common radius is a, intersect at the angle a ; find the volume common to the cylinders. The section parallel to the axes is a rhombus. i6a^ 3 sin « * 12. Find the volume generated by the revolution of one branch of the sinusoid, V = ^ sin — , a about the axis of x. n'^b ^ 2a ' 13. Find the volume enclosed by the surface generated by the revo- lution of an arc of a parabola about a chord, whose length is 2e, per- pendicular to the axis, and at a distance b from the vertex. i67r^V 15 14. Find the volume generated by the revolution of the tractrix, whose differential equation is dx -^ V{a-/) about the axis of x. Express ny^ dx in terms of y. 15. Find the volume cut from a right circular cylinder whose radius is «, by a plane passing through the centre of the base, and making the angle a with the plane of the base. 16. Find the volume generated by the curve xy^ = ^a^ (2a — x) revolving about its asymptote. 47rV J ^8 § IX.] EXAMPLES. 163 •J 17. Express the volume of a frustum of a cone in terms of its height hy and the radii ax and a^. of its bases. 7th . ^ «x — (^1 + ^1^0 + a:), 3 18. Find the volume generated by the revolution of the cardioid, r = ^ (i — cos 9), about the initial line. Express y and dx in terms of B. 8 rra^ 3 19. Find the volume of a barrel whose height is 2h, and diameter 2b, the longitudinal section through the centre being a segment of an ellipse whose foci are in the ends of the barrel. 2h' + s^'^ 27tb'^h 3 (^^ + ^0 • 20. Find the volume generated by the superior and by the inferior branch of the conchoid each revolving about the directrix ; the equation, when the axis oi y is the directrix, being xy = (^ + xy {b' - x"). n'ab' ± ^ 3 2 ,. , 4^^' Tt^ab ± 21. On two opposite lateral faces of a rectangular parallelopiped whose base is ab, oblique lines are drawn, cutting off the distances Ci, Ci, ^3, Ci on the lateral edges. A straight line intersecting each of these lines moves across the parallelopiped, remaining always parallel to the other lateral faces : find the volume cut off. ab {ci + r. + ^8 + ^i) 4 * . 22. Find the volume enclosed by the surface generated by an arc of a circle whose radius is a, about a chord whose length is 2c. 3 " 1 64 GEO ME TRICAL AP PLICA TJONS. [Ex. IX. 23. The area included between a quadrant of the ellipse A — a cos ^, y = b sm (j), and the tangents at its extremities revolves about the tangent at the extremity of the minor axis ; find the volume generated. Ttab"^ (10 — 3 tt) 24. An ellipse revolves about the tangent at the extremity of its major axis ; express the entire volume in the form of an integral, whose limits are o and 2 7r, and find its value. 211^ d^b. 25. Show that the volume between the surface, s« =:: a\x'' -^ b\y\ and any plane parallel to the plane of xy is equal to the circumscrib- ing cylinder divided by n ■\- 1. 26. A straight line of fixed length 2c moves with its extremities in two fixed perpendicular straight lines not in the same plane, and at a distance 2b. Prove that every point in the moving line describes an ellipse in a plane parallel to both the fixed lines, and find the volume enclosed by the generated surface. ^n. {c^ — Ir) b 3 27. Find the volume enclosed by the surface whose equation is x^ y^ s* Znabc a b' c' 5 28. A moving straight line, which is always perpendicular to a fixed straight line through which it passes, passes also through the circum- ference of a circle whose radius is a, in a plane parallel to the fixed straight line and at a distance b from it ; find the volume enclosed by the surface generated and the circle. na^b § IX.] EXAMPLES. 165 29. Find the volume enclosed by the surface oca and the plane a .= a. 30. Find the volume enclosed by the surface a. 2. s V x-^ -f- V* 4- s^ = rt!-\ Ttabc 2 Find Az as in Art. 107, and then evaluate V by a similar method. ^Tta^ 31. Find the volume between the coordinate planes and the surface (^,(^U^=. \a J \bj \c/ go 32. Find the volume cut from the paraboloid of revolution J* -f s' = 4ax by the right circular cylinder ^" +y = 2ax, whose axis intersects the axis of the paraboloid perpendicularly at the focus, and whose surface passes through the vertex. , i6a^ 27ta^ + . • 3 7,2i- The paraboloid of revolution .v'^ + / = cz is pierced by the right circular cylinder x^ -^r y^ = ax^ 1 66 GEOMETKICAL APPLICATIONS. [Ex. IX. whose diameter is a, and whose surface contains the axis of the parab- oloid ; find the volume between the plane of xy and the surfaces of the paraboloid and of the cylinder. 2>'^a^ 34. Find the volume cut from a sphere whose radius is ^ by a right circular cylinder whose radius is h^ and whose axis passes through the centre of the sphere. A'^V -^ , 1 zqx J~l _^,-_(^_^) J. 35. Find the volume cut from a sphere whose radius is a by the cylinder whose base is the curve r = « cos 39. 2a^7t 8^^ 3 9 -^d. Find the volume cut from a sphere whose radius is a by the cylinder whose base is the curve r = a cos B -\- sm G, z ^ 47r^^ 16 , 2 72x4 supposmg b < a. {a — y . •J y 37. A right cone, the radius of whose base is a and whose alti- tude is d, is pierced by a cylinder whose base is a circle having for diameter a radius of the base of the cone ; find the volume common to the cone and the cylinder. da^ , ^ , -(9^-16). 38. The axis of a right cone whose semi- vertical angle is a coin- cides with a diameter of the sphere whose radius is a, the vertex being on the surface of the sphere ; find the volume of the portion of the sphere which is outside of the cone. 4n'^^cos* a 3 39. Find the volume produced by the revolution of the lemniscata about a perpendicular to the initial line. Tr^a^ \/2 8~ § IX.] EXAMPLES. 167 40. Find the volumes generated by the revolution of the large loop and by one of the small loops of the curve r ^= a cos 6 cos 29 about a perpendicular to the initial line. -\ , and 16 5 32 10 * 41. From the element r dr dB dz derive the formulas for determining the volume of a solid of revolution whose axis is the axis of z. V=- 2 7t \rdrdz^ V= Tt\(r^—rl)dz, and V = 27t\(Zi — Zy)r dr. Interpret the elements in these integrals. 42. Find the volume generated by the revolution of the curve in which a > b^ about the axis oiy. Transform to polar coordinates y and use the method of Art. 139. Ttb{2b'' + 3^') na' _^ b_ 6 "^2V(«"-^')''^^ a' 43. Find the volume generated by the curve given in the preceding example, when revolving about the axis of x. na (2a' + ^b') nb" a + VC^" - b^) 1 68 GEOMETRICAL APPLICATIONS. [Ex. IX. 44. Find the volume common to the sphere whose radius is p = «, and to the solid formed by the revolution of the cardioid, r = « (i + cos 0), about the initial line. See Art. 141. — 7~ • 45. Find the whole volume enclosed by the surface Transform to the coordinates p, <^, 0, and show that the solid consists a' of four equal detached parts. -7 • X. Rectification of Plane Curves. (42. A curve is said to be rectified when Its length is deter- mined, the unit of measure to which it is referred being a right line. It is shown in Diff. Calc, Art. 314 [Abridged Ed., Art. 164], that, if s denotes the length of the arc of a curve given in rectangular coordinates, we shall have ds = ^/{dx' + df\ If the abscissas of the extremities of the arc are known, s is found by substituting for dy in this expression its value in terms of x and dx^ and integrating the result between the given values of x as limits. Thus, to express the arc measured from the vertex of the semi-cubical parabola § X.] RECTIFICATION OF PLANE CURVES. 1 69 in terms of the abscissa of its other extremity, we derive, from the equation of the curve, , 3 ^/x dx dy=- — , whence ds — — ^^^ , dx. 2 Va Integrating, = \/(gx + 4a) dx 2 f'a Jo {gx + 4ay - — . 2yVa^^ 27 (43. When x and y are given in terms of a third variable, ds is generally expressed in terms of this variable. For exam- ple, from the equations of the four-cusped hypocycloid, x — acos^ipy y = asm^(/', . . . (i) we derive dx — — -i^a cos^ ^ sin ^ d^p, and dy — ^a sin^ ^ cos ^ dip\ whence . ds = 2>^ sin tp cos d^' (2) The length of the arc between the point {a, o), corresponding to ip — o, and (o, a) corresponding to ^ = |7r, is therefore I 3^ • , r — a cos (j). . . . (3) In order to express dd in terms of ^, we substitute the value of r in (2) ; whence £n6 _i_ g-n9 ~ 2 SCC (f>, ...... {4) and by differentiation gne __ ^-ne ^ ? sec <^ tan (f> -f^ (5) w au Squaring and subtracting equation (5) from equation (4), 4 sec^ ^ r 2 . 2 JL ^^1 which reduces to M^=?^^^ (gj § X.] LENGTH OF THE LOXODROMIC CURVE. 1/3 From equations (3) and (6) dr^ — c? sin^ ^ d<^^ d^ — a^ cos^ (j) d(p ; whence substituting in the value of da (p. 171) da = aV[i +-:^)d(!>. Integrating, ff = a -^ ' d6 = a -^ — , ' (6 — a), n ]a n ^' " where a and /? denote the latitudes of the extremities of the ar^. / Examples X. I. Find the length of an arc measured from the vertex of the catenary y-^U, -' * •'■"> and show that the area between the coordinate axes and any arc is proportional to the arc. A — cs. ^ 2. Find the length of an arc measured from the vertex of the K)arabola y^ = 4ax. ^(aar f x^) -f a log -^ . J 74 GEOMETRICAL APPLICATIONS. [Ex. X. 3. Find the length of the curve €^ + I /^=- £-^ - I ' between the points whose abscissas are a and b. 1 f ^'^ — I leg V a — b, 4. Find the length, measured from the origin, of the curve y=a\og «* , a ^- X a log X, ° a — X 5. Given the differential equation of the tractrix. and, assuming (o, a) to be a point of the curve, find the value of s as measured from this point, and also the value of x in terms oi y ; that is, find the rectangular equation of the curve. y s — a log — . / ^ = tf log ^— -^-^ — V(a — y). 6. Find the length of one branch of the cycloid X— a(ip — sin ^), y = a {i — cos tp). Sa. 7. When the cycloid is referred to its vertex, the equations being X — a {i — cosip)f ^ = « (^ + sin ?/?), prove that s = \/{Sax). § X.] EXAMPLES. i;5 8. Find the length from the point (^, o) of the curve X ^= 2a cos ip — a cos 2tp, _y = 2asmtp — a sin 2tp. 4a(tp — sin tp). 9. Show that the curve, X = ^a cos 'Z' — 2a cos* ^, y= 2a sin' ^^, has cusps at the points given by i/^ = o and tp = tt ; and find the whole length of the curve. 12a. 10. Find the length of a quadrant of the curve See Ftg. 6, Art. 107. -— - , a ^r o II. Show that the curve x=^ 2a cos* ^ (3 — 2 cos^ ^), j; = 4^ sin ^ cos^ ^ has three cusps, and that the length of each branch is — . / J 12. Find the length of the arc between the points at which the curve A' = ^cos'^ ^cos2^, _;' =«sin^^sin 26^ 2-i/2 cuts the axes. . a. ' — r~ ^^^ .;^ GEOMETRICAL APPLICATIONS. [Ex. X. N 13. 3how that the curve \A*^ xJ^ ^ ^ — (^ cos ^ (i + sin" ^), ^X y = a^xntp cos^* ^ is symmetrical to the axes, and find the length of the arcs between the cusps. / , . I ^ • a [ \/2 — sm-^ — \ V3 a[ 4/2 '+ cos"^ — - J . \ 4/3/ \ 14. Find the length of one branch of the epicycloid / 7v / 7 a -\r b . X = [a -i- P) cos ip — o cos — 7 — ^, ^ = (^ + ^) sin ^ — ^ sin — - — ^. U {a + b) a 15. Show that the curve X = ga sin tp — 4a sin' ^, y = — 3 cos Q = , and sm a = . tan y tdiVi y Squaring and adding, _ tan^ ip + tan^ (f> ~~ tan^ y ' or tan^ y = tan*^ tf) + tan^ (p ; whence sec' y=i+ (J^)' + (|)'. Substituting in the formula derived in Art. (152), we have 154. It is sometimes more convenient to employ the polar 1 82 GEOMETRICAL APPLICATIONS. [Art. 1 54. element of the projected area. Thus the formula becomes 5 = sec yrdrdOy where sec y has the same meaning as before. For example, let it be required to find the area of the sur- face of a hemisphere intercepted by a right cylinder having a radius of the hemisphere for one of its diameters. From the equation of the sphere, x^+f + ^=d\ (i) we derive dz _ X dz _ y dx z^ dy z * whence --=^[■^©•-(1) therefore * 5 ' ' -«jf the integration extending over the area of the circle r = a cos 6 (2) Since equation (i) is equivalent to ^ + r2 = ^, § XI.] AREAS OF SURFACES IN GENERAL. 1 83 From (2) the limits for ;■ are ri = o, and r^ — a cos ^, hence 5r=^|(i -sin6')^/9, in which a sin Q is put for \.\\^ positive quantity s/(c? — r}). The limits for Q are —\'n: and \7i^ but since sin /9 is in this case to be regarded as invariable in sign, we must write 5 ■= 2^2 [ '(I _ sin B) dO = na^ - 2a\ Jo If another cylinder be constructed, having the opposite radius of the hemisphere for diameter, the surface removed is 27ta^ — 4a^, and the surface which remains is 4^?^ a quantity commensurable with the square of the radius. This problem was proposed in 1692, in the form of an enigma, by Vivian i, a Florentine mathematician. Examples XL I. Find the surface of the paraboloid whose altitude is a-:, and the radius of whose base is d. 2. Prove that the surface generated by the arc of the catenary given in Ex. X., I, revolving about the axis of x, is equal to 7t{cx + sy). 3. Find the whole surface of the oblate spheroid produced by the 84 GEOMETRICAL APPLICATIONS. [Ex. XI. revolution of an ellipse about its minor axis, a denoting the major, b. the minor semi-axis, and e the excentricity, — ^^ . 2 . b\ 1 +e 27ta -i- TV - log -. 4. Find the whole surface of the prolate spheroid produced by the revolution of the ellipse about its major axis, using the same notation as in Ex. 3. ,2 , -sin"^^ 2 7to -}- 2 7rab . 5. Find the surface generated by the cycloid X z= a {(p — sin //?), y = a {i — cos //') revolving about its base. — 7r«^ 3 6. Find the surface generated when the cycloid revolves about the tangent at its vertex. 7. Find the surface generated when the cycloid revolves about its axis. 8. Find the surface generated by the revolution of one branch of the tractrix (see Ex. X., 5) about its asymptote. § XL] EXAMPLES. 185 9. Find the surface generated by the revolution about the axis of X of the portion of the curve y = f ^ which is on the left of the axis of y. 7r[ |/2 + log (i + V2)]. 10. Find the surface generated by the revolution about the axis of X of the arc between the points for which x = a and :xr = ^ in the hyperbola xy = k^. II. Show that the surface of a cylinder whose generating lines are parallel to the axis of z is represented by the integral 5 = \z t/Sj where s denotes the arc of the base in the plane of xy. Hence, deduce the surface cut from a right circular cylinder whose radius is a, by a plane passing through the centre and making the angle ^ with the plane of the base. za^ tan a. 12. Find the surface of that portion of the cylinder in the problem solved in Art. 154, which is within the hemisphere. 20^. 13. Find the surface of a circular spindle, a being the radius and 2c the chord. 47ra c- t/(a'^-^=^jsin-- 1 86 GEOMETRICAL APPLICATIONS. [Art. 1 5 5, XII. The Area generated by a Straight Line moving in any Manner in a Plane, 155. If a straight line of indefinite length moves in any man- ner whatever in a plane, there is at each instant a point of the line about which it may be regarded as rotating. This point we shall call the centre of rotation for the instant. The rate of motion of every point of the line in a direction perpendic- ular to the line itself is at the instant the same as it would be if the line were rotating at the same angular rate about this point as a fixed centre."^ Hence it follows that the area generated by a definite portion of the line has at the instant the same rate as if the line were rotating about a fixed instead of a variable centre. (56. Suppose at first that the centre of rotation is on the generating line produced, p^ and p^ denoting the distances from the centre of the extremities of the generating line, and let denote its inclination to a fixed line. By substitution in the general formula derived in Art. no, we have dA =Yp}-p^)cU. ♦Compare Difif. Calc, Art. 332 [Abridged Ed., Art. 176J, where the moving line is the normal to a given curve, and the centre of rotation is the centre of cur- vature of the given curve. If the line is moving without change of direction, the centre is of course at an infinite distance. When the line is regarded as forming a part of a rigidly connected system in motion, its centre of rotation is the foot of a perpendicular dropped upon it from the instantaneous centre of the motion of the system. Thus, if the tangent and normal in the illustration cited are rigidly connected, the centre of curvature, C, is the instantaneous centre of the motion of the system, and the point of contact, P, is the centre of rotation for the tangent line. § XII.] AREAS GENERATED BY MOVING LINES. 1 8/ Applications, 157. The area between a curve and its evolute may be generated by the radius of curvature p, whose inclination to the axis of ;ir is ^ + ^n^ in which (j) denotes the inclination of the tangent line. Since the centre of rotation is one extremity of the generating linep, the differential of this area is found by substituting in the general expression Pi = o and P2 = o. Hence when p is expressed in terms of ^, ^ = Mp2^0 expresses the area between an arc of a given curve, its evolute, and the radii of curvature of its extremities, the limits being the values of (t> at the ends of the given arc. 158. For example, in the case of the cardioid r — a{i — cos 6), it is readily shown, from the results obtained in Art. 145, that the angle between the tangent and the radius vector is ^6\ and therefore ^ = |(9, and ds Aa . 6 To obtain the whole area between the curve and its evolute, the limits for B are o and 27t ; hence the limits for ^ are o and 37r. Therefore A '%^d^=^J^r^r.^td^ = ^^, 159. As another application of the general formula of Art. 156, let one end of a line of fixed length a be moved 1 88 GEOMETRICAL APPLICATIONS, [Art. 1 59. along a given line in a horizontal plane, while a weight at- tached to the other extremity is drawn over the plane by the line, and is therefore always moving in the direction of the line itself. The line of fixed length in this case turns about the weight as a moving centre of rotation. Hence the area generated while the line turns through a given angle is the same as that of the corresponding sector of a circle whose radius is a. The curve described by the weight is called a tractrix^ and the line along which the other extremity is moved is the direc- trix. When the axis of x is the directrix, and the weight starts from the point (o, a), the common tractrix is described ; hence the area between this curve and the axis is ^na^. 160. Again, in the generation of the cycloid, Diff. Calc, Art. 288 [Abridged Ed., Art. 156], the variable chord RP may be regarded as generating the area. The point R has a motion in the direction of the tangent RX\ the point P partakes of this motion, which is the motion of the centre Cj and also has an equal motion, due to the rotation of the circle in the direc- tion of the tangent to the circle at P. Since the tangents at P and R are equally inclined to PRy the motion of P in a direction perpendicular to PR is double the component, in this direction, of the motion of R, Therefore the centre of rota- tion of PR is beyond i? at a distance from it equal to PR, Hence, denoting PRO by ^, Pi — PR — 2a sin <^, pg = '2-PR = 4^ sin ^. Substituting in the formula of Art. 156, we have for the area of the cycloid, since PRO varies from o to n^ A = 6^^ sin^ ^ d — -^nc?. § XIL] SIGN OF THE GENERATED AREA. 189 Sign of the Generated Area. (61. Let AB be the generating line, and ^ the centre ol rotation. The expression, dA = i {Pi - P?) d(l>, (0 Fig. ig. for the differential of the area, was obtained upon the supposi- tion that A and B were on the same side of C. Then suppos- ing P2 > Pi, and that the line rotates in the positive direction^ as in figure 19, the differential of the area is positive; and we notice that every point in the area generated is swept over by the line AB, the left hand side as we face in the direction A B preceding. 162. We shall now show that in every case, the formula requires that an area swept over with the left side preceding, shall be considered as positively generated, and one swept over in the opposite direction as negatively generated. In the first place, if C is between A and B so that p, is negative, as in figure 20, p^ is still positive, and formula (i) still gives the difference between the areas generated hy AB and AC. Hence the latter area, which is now generated by a part of the line AB, must be regarded as generated negatively, but the right hand side as we face in the direction AB of this portion of the line is now preceding, which agrees with the rule given in Art. 161. Again, if C is beyond B, the formula gives the difference of the generated areas ; but since pi is numerically greater than p2^, in this case, dA is negative, and the area generated by AB is the difference of the areas, and is negative by the rule. Fig. 20. 190 GEOMETRICAL APPLICATIONS. [Art. 162. Finally, if the direction of rotation be reversed, d(l> and therefore dA change sign, but the opposite side of each por- tion of the line becomes in this case the preceding side. 163. We may now put the expression for the area in another form. For dA=~{pl-ri)d^ (Ps-pO^^V^; whatever be the signs of Pg and Pi, the first factor is the length of AB, which we shall denote by /, and the second factor is the distance of the middle point of AB from the centre of rotation, which we shall denote by p^„. Hence, putting P2 - Pi = /, we have and Ipm d(l>. P2+ Px__ (2) Since p^ d(j> is the differential of the motion of the middle point in a direction perpendicular to AB, this expression shows that the differential of the area is the product of this differential by the length of the generating line. Areas generated by Lines whose Extremities describe Closed Circuits. i-^ I64-. Let us now suppose the generating line AB to move from a given position, and to return to the same position, each of the extremities A and B describing a closed curve in the positive direction, as indicated by the arrows in figure 21. It is readily seen that every point which is in the area described by B, and not in that described by A^ will be swept over at least once by the line AB^ the left side preceding, Fig. 21. and if passed over more than once, there will be § XII.] AREAS GENERATED BY MOVING LINES, I9I an excess of one passage, the left side preceding. Therefore the area within the curve described by i?, and not within that described hy A, will be generated positively. In like manner the area within the curve described by A, and not within that described by B, will be generated negatively. Furthermore, all points within both or neither of these curves are passed over, if at all, an equal number of times in each direction, so that the area common to the two curves and exterior to both disap- pears from the expression for the area generated by AB. Hence it follows that, regarding a closed area whose perimeter is described in the positive direction as positive^ the area generated by a line returning to its original position is the difference of the areas described by its extremities. This theorem is evidently true generally, if areas described in the opposite direction are regarded as negative. Amslers Planimeter, 165. The theorem established in the preceding article may be used to demonstrate the correctness of the method by which an area is measured by means of the Polar Planimeter, invented by Professor Amsler, of Schaffhausen. This instrument consists of two bars, OA and AB, Fig. 22, jointed together at A, The rod OA turns on a fixed pivot at (9, while a tracer at B is carried in the positive direction completely around the perimeter of the area to be measured. At some point C of the bar AB a small wheel is fixed, having its axis parallel to AB, and its circumference resting upon the paper. When ^is moved, this wheel has a sliding and a roll- ing motion ; the latter motion is recorded by an attachment by means of which the number Fig. 22. of turns and parts of a turn of the wheel are registered. 192 GEOMETRICAL APPLICATIONS. [Art. 166. 166. Let J/ be the middle point of AB, and let OA =a, AB = b, MC^c. Since b is constant, the area described hy AB is by equation (2), Art. 163, \9md(l> (l) Kx^2.AB — b Denoting the linear distance registered on the circumference of the wheel by s^ ds is the differential of the motion of the point C, in a direction perpendicular to AB^ and since the dis- tance of this point from the centre of rotation is Pni + ^, ds = {p„t + c) d(f) : substituting in (i) the value of pmd(l>y ArGSiAB = b{ds-bAd^ (2) 167. Two cases arise in the use of the instrument. When, as represented in Fig. 22, O is outside the area to be meas- ured, the point A describes no area, and by the theorem of Art. 164, equation (2) represents simply the area described by B. In this case ^ returns to its original value, hence d(f> vanishes, and denoting the area to be measured by^, equation (2) becomes A = 6s (3) In the second case, when O is within the curve traced by By the point A describes a circle whose area is Ttd^j and the limit- § XII.] AMSLER'S PLANIMETER. 193 ing values of differ by a complete revolution. Hence in this case equation (2) becomes A — ncP- — bs — 2 Ttbcy or A= bs ^- 7c{c? — 2bc)!^ (4) In another form of the planimeter the point A moves in a straight line, and the same demonstration shows that the area is always equal to bs. Examples XII. I. The involute of a circle whose radius is a is drawn, and a tangent is drawn at the opposite end of the diameter which passes through the cusp ; find the area between the tangent and the involute. a'n (3 + n'') 2. Two radii vectoresof a closed oval are drawn from a fixed point within, one of which is parallel to the tangent at the extremity of the other ; if the parallelogram be completed, the area of the locus of its vertex is double the area of the given oval. 3. Show that the area of the locus of the middle point of the chord joining the extremities of the radii vectores in Ex. 2, is one half the area of the given oval. * The planimeter is usually so constructed that the positive direction of rotation is with the hands of a watch. The bar b is adjustable, but the distance y^ C is fixed so that c varies with b. Denoting AChy q, we have c = q — \b, and the constant to be added becomes C =^ it {a^ — ibq -\- b'^) in which a and^ are fixed and b adjusta- ble. In some instruments q is negative. It is to be noticed that in the second case s may be negative ; the area is then the numerical difiference between the constant and bs. 194 GEOMETRICAL APPLICATIONS. [Ex. XII. 4. Prove that the difference of the perimeters of two parallel ovals, whose distance is 3, is 2 nb, and that the difference of their areas is the product of b and the half sum of their perimeters. 5. A lima9on is formed by taking a fixed distance be on the radius vector from a point on the circumference of a circle whose radius is a ; show that the area generated by b when b'> 2« is the area of the lima- 9on diminished by twice the area of the circle, and thence determine the area of the lima9on. 7r(2^^ + b''), 6. Verify equation (4), Art. 167, when the tracer describes the circle whose radius \^ a -\- b. 7. Verify the value of the constant in equation (4), Art. 167, by determining the circle which may be described by the tracer without motion of the wheel. 8. If, in the motion of a crank and connecting rod (the line of motion of the piston passing through the centre of the crank), Amsler's record- ing wheel be attached to the connecting rod at the piston end, deter- mine s geometrically, and verify by means of the area described by the other end of the rod. 9. The length of the crank in Ex, 8 being a^ and that of the con- necting rod b, find the area of the locus of a point on the connecting rod at a distance c from the piston end. 10. If a line AB of fixed length move in a plane, returning to its original position without making a complete revolution^ denoting the areas of the curves described by its extremities by {A) and (^), determine the area of the curve described by a point cutting AB in the ratio m : n. n{A) 4- m(B ) m + n § XII.] EXAMPLES. 195 II. If the line in Ex. 10 return to its original position after making a complete revolution^ prove Holditch's Theorem j namely, that the area of the curve described by a point at the distance c and c from A and B is c'{A) + c(B) c ■{- c' ncc 12. Show by means of Ex. 11 that, if a chord of fixed length move around an oval, and a curve be described by a point at the distances c and c from its ends, the area between the curves will be ttcc . XIII. Approximate Expressions for Areas and Volumes. 168. When the equation of a curve is unknown, the area between the curve, the axis of x, and two ordinates may be approximately ex- pressed in terms of the base and a lim- ited number of ordinates, which are sup- posed to have been measured. Let ABCDE be the area to be de- termined ; denote the length of the base by 2h ; and let the ordinates at the ex- tremities and middle point of the base be measured and denoted by y^.y^, and jj/g. Taking the base for the axis of x, and the middle point as origin, let it be assumed that the curve has an equation of the form y^ A ^ Ex ^ C^-V D:^ ; (i) Fig. 23. then the area required is ^ V' ^ ^ Bj^ C^ DxT A — ydx-Ax^- — H + J-/ 2 3 4 _ in which which A and C are unknown '' ^-{(yA^2Ch% . (2) -h 3 ig6 GEOMETRICAL APPLICATIONS. [Art. i68. In order to express the area in terms of the measured ordinates, we have from equation (i), whence we derive and substituting in (2), A {yi + 4j2 + J/3). It will be noticed that this formula gives a perfectly ac- curate result when the curve is really a parabolic curve of the third or a lower degree. 169. If the base be divided into three equal intervals, each denoted by k, and the ordinates at the extremities and at the points of division measured, we have, by assuming the same equation, A=\^\^jdx=^-^{AA + iOf) (I) From the equation of the curve, y,=A Fig. 24. 3^^ gCh^ 27Dh^ , Bh Ch^ Dh^ y,^A-^^----^- , Bh Ch^ Dk^ ^4 ^ 4. 3^ 4- 9^' + ?Z^' ; § XIII.] SIMPSON'S RULES. 1 97 + 9f^. whence ji + }\ = 2 A From these equations we obtain i6 and a^=^^--'^^-^A^V 4 Substituting in equation (i), A =^iyi+ 3J2+ 3/8 +J4). Simpson s Rules. !70. The formulas derived in Articles 168 and 169, although they were first given by Cotes and Newton, are usually known as Simpsons Rules, the following extensions of the formulas having been published in 1743, in his Mathematical Disserta- tions. If the whole base be divided into an even number n of parts, each equal to h, and the ordinates at the points of divis- ion be numbered in order from end to end, then by applying the first formula to the areas between the alternate ordinates, we have That is to say, the area is equal to the product of the sum of the extreme ordinates, four times the sum of the even-num- 198 GEOMETRICAL APPLICATIONS. [Art. I7O0 bered ordinates, and twice the sum of the remaining odd-num- bered ordinates, multipHed by one third of the common interval. Again, if the base be divided into a number of parts divis- ible by three, we have, by applying the formula derived in Art. 169, to the areas between the ordinates ^^1:^4,^4/7, and so on, ^ "^ V ^^' "^ ^-^^ "^ ^^^ + 274 + 3J/5 . . . + 3J« + Jn+x). Cotes Method of Approximation, 171. The method employed in Articles 168 and 169 is known as Cotes Method. It consists in assuming the given curve to be a parabolic curve of the highest order which can be made to pass through the extremities of a series of equi- distant measured ordinates. The equation of the parabolic curve of the ;^th order con- tains n ■\- \ unknown constants; hence, in order to eliminate these constants from the expression for an area defined by the curve, it is in general necessary to have n + i equations con- necting them with the measured ordinates. Hence, if n de- note the number of intervals between measured ordinates over which the curve extends, the curve will in general be of the n\\\ degree.* * \i H denotes the whole base, the first factor is always equivalent to H divided by the sum of the coefficients of the ordinates ; for if all the ordinates are made equal, the expression must reduce to Hy^. Thus, each of the rules for an approximate area, including those derived by repeated applications, as in Art. 170, may be regarded as giving an expression for the mean ordinate. The coefficients of the ordinates, according to Cotes' method, for all values of « up to w = 10, may be found in Bertrand's Calcul Integral, pages 333 and 334. For example (using detached coefficients for brevity), we have, when « = 4, -4 = — [7, 32, 12, 32, 7] ; and when « = 6, TT A ■= - — [41, 216, 27, 272, 27, 216, 41]. 040 § XIII.] THE FIVE-EIGHT RULE. I99 172. For example, let it be required to determine the area between the ordinates yi and j/2, in terms of the three equi- distant ordinates/i, Jo and_)/3, the common interval being h. We must assume y= A ^- Bx -v C^\ then taking the origin at the foot of ji, A=\yd.= hlA^-^-\^ from which A, B and C must be eliminated by means of the equations yQ = A + 2Bh + ^Cl^, Solving these equations, we obtain 2 If we make a slight modification in the ratios of these last coefficients by sub*, stituting for each the nearest multiple of 42, we have A — - — [42, 210, 42, 252, 42, 210, 42], 840 (the denominator remaining unchanged, since the sum of the coefficients is still 840), which reduces to IT ^ = — [i, 5, I, 6, I, 5, ij. This result is known as Weddles Rule for six intervals. The vallie thus given to the mean ordinate is evidently a very close approximation to that resulting, from Cotes' method, the difference being 840 l^^i + ■^" + ^5 {y-i + Jo) - 6 {yi + jKe) - 20^4]. 200 GEOMETRICAL APPLICATIONS. [Art. 1 72. and substituting h 173. It is, however, to be noticed, that when the ordinates are symmetrically situated with respect to the area, if n is everiy the parabolic curve may be assumed of the {n + i)th degree. For example, in Art. 168, 71 — 2, but the curve was assumed of the third degree. Inasmuch as A, B, C and D cannot all be expressed in terms of j/j, y^, and y^, we see that a variety of parabolic curves of the third degree can be passed through the extremities of the measured ordinates, but all of these curves have the same area."^ Application to Solids, (74. \i y denotes the area of the section of a solid perpen- dicular to the axis of x, the volume of the solid is \ydx, and * This circumstance indicates a probable advantage in making n an even num- ber when repeated applications of the rules are made. Thus, in the case of six intervals, we can make three applications of Simpson's first rule, giving TT A = - - [i, 4, 2, 4, 2, 4, i], (i) 10 or two of Simpson's second rule, giving '■^ = ^ [i, 3, 3, 2, 3, 3, i] (2) In the first case, we assume the curve to consist of three arcs of the third degree, meeting at the extremities of the ordinates ^3 and jr, ; but, since each of these arcs contains an undetermined constant, we can assume them to have common tangents at the points of meeting. We have therefore a smooth, though not' a continuous curve. In the second case, we have two arcs of the third degree containing no arbitrary constants, and therefore making an angle at the extremity of jj/4. It is probable, therefore, that the smooth curve of the first case will in most cases form a better approximation than the broken curve of the second case. In confirmation of this conclusion, it will be noticed that the ratios of the coefficients in equation (i) are nearer to those of Cotes' coefficients for « = 6, given in the preceding foot-note, than are those in equation (2). § XIII.l APPLICATION TO SOLIDS. 20I therefore the approximate rules deduced in the preceding arti- cles apply to solids as well as to areas. Indeed, they may be applied to the approximate computation of any integral, by putting J/ equal to the coefficient of x under the integral sign. The areas of the sections may of course be computed by the approximate rules. Woolley's Rule, 175. When the base of the solid is rectangular, and the ordinates of the sections necessary to the application of Simp- son's first rule are measured, we may, instead of applying that rule, introduce the ordinates directly into the expression for the area in the following manner. Taking the plane of the base for the plane of xy^ and its centre for the origin, let the equation of the upper surface be assumed of the form z=A ^Bx^Cy^D.^^Exy^Ff-vGj(^^Hx''y\-Ixf^Jf. Let 2h and 2k be the dimensions of the base, and denote the measured values of z as indicated in Fig. 25. The required volume is '^ = 1 \ ^dy ) -h i -k dx. This double integral vanishes for every term containing an odd power of x or an odd power oi y\ hence hh = — [12^ + 4i;>^ + 4^>^]. (I) 202 GEOMETRICAL APPLICATIONS. [Art. 1 75. By substituting the values of x and y in the equation of the surface, we readily obtain b2 = A, (2) ^1 + <3:3 + ^1 + ^s = 4^ + 4Dh^ + aF]^, ... (3) «2 + ^2 + '^i + <^3 = 4^ + '2'Dh^ + 2FB, ... (4) From these equations two very simple expressions for the volume may be derived ; for, employing (2) and (4), equation (i) becomes ^=^(^ + ^1 + 2^2 + ^3+^2); . . . . (4) and employing (2) and (3), hk F= — - (^1 + ^ + 8/^2 + <^i + ^^s) (5) Equation (4) is known as Woolleys Rule ; the ordinates employed are those at the middles of the sides and at the centre ; in (5), they are at the corners and at the centre. Examples XI 11. 1. Apply Simpson's Rule to the sphere, the hemisphere, and the cone, and explain why the results are perfectly accurate. 2. Apply Simpson's Second Rule to the larger segment of a sphere made by a plane bisecting at right angles a radius of the sphere. ~8~' § XIII.] EXAMPLES. 203 3. Find by Simpson's Rule the volume of a segment of a sphere, b and c being the radii of the bases, and h the altitude. 4. Find by Simpson's Rule the volume of the frustum of a cone, b and c being the radii of the bases, and h the altitude. — {lfi^-bc-\- c"). 5. Compute by Simpson's First and Second Rules, the value of , the common interval being ^V in each case. oi + ^ ^ ^'' The first rule gives 0.6931487, and the second rule gives 0.6931505. The correct value is obviously loge2 = 0.6931472. 6. Find the volume considered in Art. 175, directly by Simpson's Rule, and show that the result is consistent with equations (4) and (5). hk y= — [ai -{- as -h Ci + d -{- 4 {a d(j)y - 2V2 f 2 ., ,, ^ , ^ J . .. X =■ a cos*(p d(f) — (7 = -TT- Tta. 3 •'o 24/2 3-1 TV _ V2 Solids of Revolution, 185. To find the centre of gravity of a solid of revolution, we take the axis of revolution as the axis of x^ and the circle whose area is nf' as the generating element. Replacing y in equation (3), Art. 178, by this expression, we have for the stati- cal moment 7t xf" dx, and for the abscissa of the centre of gravity _ xfdx X — ^ "^ dx i a 210 MECHANICAL APPLICATIONS. [Art. 1 86. 186. To illustrate, we find the centre of gravity of a spheri- cal segment whose height is //. In this case, taking the origin at the vertex of the segment, and denoting the radius of the sphere by a^ we have f- — 2ax — :^. fh 2 I "l'^ {2ax' — ^) dx -a^ - -x^ \ , ^ Hence x = h ^ 3 A ^q ^h ^a - ^h ^ f' (2ax - x^) dx a^ - -^t-^T 4 3^-^* Jo 3 Jo If the centre of gravity of the surface of the segment be re- quired, since the differential of the surface is 27ty ds, we easily obtain the general formula x = and, in this case the curve being a circle, y ds = a dx] hence, substituting, we have X = ^h. The Properties of Pappus, 187. Let a solid be generated by the revolution of any plane figure about an exterior axis in its own plane. It is required to determine the volume and the surface thus generated. It is evident that this solid may also be generated by a variable circular ring whose centre moves along the axis of revolution ; denoting by jj and 72 corresponding ordinates of § XIV.] THE PROPERTIES OF PAPPUS. 211 the outer and inner circles respectively, the area of this ring is 7i{yi — yi). Hence But this integral is the statical moment of the given figure, since /i — y^ is the generating element of its area, and — — ^is the, corresponding arm. Denoting the area of the figure by Ay we may therefore write V= 2nyA ; that is, the volume is the product of the area of the figure and the path described by its centre of gravity. The surface (5) of this solid is, by Art. 149, S — 27t\yds =27t\dSf if J denotes the ordinate of the centre of gravity of the arc s. Hence we have S= ZTrj-arc ; that is, the surface is the product of the length of the arc into the path described by the centre of gravity. These theorems are frequently called the properties of Gul- dinus ; they are, however, due to Pappus, who published them 1588. It is obvious that both theorems are true for any part of a revolution of the generating figure. 212 MECHANICAL APPLICATIONS. [Ex. XIV, Examples XIV. 1. Find the centre of gravity of the area enclosed between the parabola y^ = ^mx and the double ordinate corresponding to the abscissa a. 5 2. Find the centre of gravity of the area between the semi-cubical parabola af = x^ and the double ordinate which corresponds to the abscissa a. 7 3. Find the ordinate of the centre of gravity of the area between the axis of x and the sinusoid y = sin .r, the limits being x = o and x=7t. y=i7r. 4. Find the coordinates of the centre of gravity of the area be- tween the axes and the parabola ey-©*- X = — , and y = - 5 5 5. Find the centre of gravity of the area between the cissoid f {a — x) — x^ and its asymptote. Solution : — Denoting the statical moment by M and the area by A, M = = — 2x-^ (a — xY +5 ^^"^ {^ — -^'f^ ^x Jo {a — X)-'- Jo Jo = z^a- A- sM; ,'.M-=^A, hence a=^. o § XIV.] EXAMPLES. 213 6. Find the centre of gravity of the area between the parabola v' = ^ax and the straight line j ~ mx. — %a . - 211 X ■= — :, , and y — — . 5w m 7. Find the centre of gravity of the segment of an ellipse cut off by a quadrantal chord. — 2a , — 2 b X = - • , and y $ 7r — 2 - s TT- 2 8. Given the cycloid, y — a{i — cos ?/.'), X = a (ip — sin ip) , find the distance of its centre of gravity from the base. ^ = 6- 9. Find the centre of gravity of the area enclosed between the positive directions of the coordinate axes and the four-cusped hypo- cycloid x^ -\- y^ = a^. Put .\" = « cos^ 0, and y = a sin' 0. - 3^^ x=y 10. Find the centre of gravity of the area enclosed by the cardioid . = a(i — cos 0). ^=-f II. Find the centre of gravity of the sector of a circle whose radius is a, the angle of the sector being 2 a. — 2 a sin (X Use the method of Art. i^2>- '*' 3 a 214 MECHANICAL APPLICATIONS. [Ex. XIV. 12. Find the centre of gravity of the segment of a circle, the angle subtended being 2 a and the radius of the circle a. Solution X J a cos a 2 1 \a —X ) xdx 3.3 3 2a sin a Chord Area 3 Area 12 Area 13. Find the centre of gravity of a circular ring, the radii being a and «i, and the angle subtended 2a. - _ 2 d — a^ sin (y 3 d^ — a-c oc 14. Find the centre of gravity of a circular arc, whose length is 2s. Soliction : — We have in this case, taking the origin at the centre and the axis of X bisecting the arc, xds X ds Put X — a cos 0, then ds — a dB, and denoting by a the angle subtended by ^, we have fOL X = COS do ^ sin « c 2s a a 2c being the chord. § XIV.] EXAMPLES. 215 15. Find the coordinates of the centre of gravity of arc of the semi- cycloid whose equations, referred to the vertex, are ;r = « (i — cos ^'), and j == « (^ + sin ^). ^^, andj^r^ [n-Yja- X 16. Find the centre of gravity of the arc between two successive cusps of the four-cusped hypocycloid x^ 4-7^ = a^\ _ _ _ _ 2d! 17. Find the position of the centre of gravity of the arc of the semi- cardioid r = « (i — cos 6). x= , and y = — . 18. A semi-ellipsoid is formed by the revolution of a semi-ellipse about its major axis ; find the distance of the centre of gravity of the solid from the centre of the ellipse. x-^ 19. Find the centre of gravity of a frustum of a paraboloid of revolution having a single base,, k denoting the height of the frustum. '*^~ 3 ■ 20. A paraboloid and a cone have a common base and vertices at the same point ; find the centre of gravity of the solid enclosed between them. The centre of gravity is the middle point of the axis. 2l6 MECHANICAL APPLICATIONS, [Ex. XIV. 21. Find the centre of gravity of a hyperboloid whose height is hy the generating curve being y^ — m (2ax + ^'). — k Sa + s^ x = V . 4 s^-\- k 22. Find the centre of gravity of the solid formed by the revolution of the sector of a circle about one of its extreme radii. The height of the cone being denoted by ^, and the radius of the circle by «, we have 23. Find the centre of gravity of the solid formed by the revolution about the axis of x of the curve ay = ax^ — x^f between the limits o and a. x-^ 24. A solid is formed by revolving about its axis the cardioid r — a (i — cosG) ; find the distance of the cusp from the centre of gravity. — _ i6a 25. Determine the position of the centre of gravity of the volume included between the surfaces generated by revolving about the axis of .;*: the two parabolas y = mxy and y^ = m' {a — x). - a m + 2m' X 3 m + m § XIV.] EXAMPLES, 217 26. Find the centre of gravity of a rifle bullet consisting of a cylin- der two calibers in length, and a paraboloid one and a half calibers in length having a common base, the opposite end of the cylinder con- taining a conical cavity one caliber in depth with a base equal in size to that of the cylinder. The distance of the centre of gravity from the base of the bullet is if I calibers. 27. A solid formed by the revolution of a circular segment about its chord is cut in halves by a plane perpendicular to the chord ; determine the centre of gravity of one of the halves. This solid is called an ogival. Denoting hy 2a the angle subtended by the chord, and by a the radius of the circle, the distance of the centre of gravity from the base is - _ a 44 sin*^ a + sin' 2a + 32 (cos 201 — cos a) X = 6 sin Of (2 -I- cos" a) — ^a cos a 28. Find the centre of gravity of the surface of the paraboloid formed by the revolution about the axis of x of the parabola / = 4mx, a denoting the height of the paraboloid. - _ I (3^ ~ 2»2) {a + m)^ -t- 2fn^ X — — • — — , 5 {a -\- my — m^ 29. Find the centre of gravity of the surface generated by the revo- lution of a semi-cycloid about its axis, the equations of the curve being ;i; = ^(i _ cos ^), and jj' = d! (^ + sin ^). - 2a IKTT — 8 x = ^ . 15 3^-4 2l8 MECHANICAL APPLICATIONS. [Ex. XIV. 30. Find the centre of gravity of the surface generated by the revo- lution about its axis of one of the loops of the lemniscata r^ =1 a^ cos 20. - 2 + V2 X :=■ — -— a. 31. A cardioid revolves about its axis ; find the centre of gravity of the surface generated, the equation of the cardioid being r = a {1 — cos9)- 63 — f^oa 32. A ring is generated by the revolution of a circle about an axis in its own plane ; c being the distance of the centre of the circle from the axis, and a the radius, determine the volume and surface generated. V— 27t^cc^^ and S— ^n'^ca. 33. A triangle revolves about an axis in its plane ; ax, a^, and a^^ denoting the distances of its vertices from the axis, determine the vol- ume generated. 271 A , . V — ■ \ax + ^2 + ^3). 34. Find the Volume of a frustum of a cone, the radii of the bases being ax and a^^, and the height h, 7th {ux + axa^ + «/). 35. Find the volume and surface generated by the revolution of a cycloid about its base. 647ra^ V= 57rV, and S = § XV.] MOMENTS OF INERTIA, 2ig XV. Moments of Inertia, 188. When a body rotates about a fixed axis, the velocity of a particle at a distance r from the axis is in which go is the angle of rotation. The force which acting for a unit of time would produce this motion in a mass m is measured by the momentum daa mr —r- . The moment of this force about the axis is therefore o ddj m'T —T" dt The sum of these moments for all the parts of a rigid system is since the angular velocity, -5- , is constant. In the case of a dt continuous body this expression becomes in which dm is the differential of the mass. The factor \r^dm^ 220 MECHANICAL APPLICATIONS. [Art. 1 88. which depends upon the shape of the body, is called its mo^ ment of inertia^ and is denoted by /. 189. When the body is homogeneous, dm is to be taken equal to the differential of the line, area, or volume, as the case may be. For example, in finding the moment of inertia of a straight line whose length is 2a^ about an axis bisecting it at right angles, we let x denote the distance of any point from the axis; then dm = dx, hence we have I^l' x^dx = ^-^=^^ J-. ^ 12 Again, in finding the moment of inertia of the semi-circle in figure 25, about the axis of j^, let dm= 2ydx\ then, since every point of the generating line is at the distance x from the axis, the moment of inertia is 7=2 yx^ dx = 2\ V{a^ — ^) ^^ dx , Jo Jo Putting X — a sin 6, we have 1= 20^ f' cos^ e sin2 ede = ^\ Jo O T/^e Radius of Gyration, 190. If the whole mass of the body were situated at the distance k from the axis, its moment of inertia would be Bm. Now, if k is so determined that tJns moment shall be equal to the actual moment of inertia of the body, the value of k is the radius of gyration of the body with reference to the given axis. Hence j^ __ Moment of inertia Mass * § XV.] THE RADIUS OF GYRATION. 221 Thus, for the radius of gyration of the line 2a^ whose moment of inertia is found in the preceding article, we have >r=- , or k= —\ 3 V3 and for the radius of gyration of the semi-circle, whose area is \7ia^y J^^""-, or k^""-. 4 2 It is evident that this expression is also the radius of gyra- tion of the whole circle about a diameter, for the moment of inertia of the circle is evidently double that of the semi-circle, and its area is also double that of the semi-circle. 191. It is sometimes convenient to use modes of generating the area or volume, other than those involving rectangular coordinates. For example, let it be required to find the radius of gyration of a circle whose radius is a^ about an axis passing through its centre and perpendicular to its plane. This circle may be generated by the circumference of a variable circle whose radius is r, while r passes from o to a. The differential of the area is then 2nr dr, and the moment is I = 27t\ 7^ dr = — , 'i: Dividing by the area of the circle, we have 192. Again, to find the radius of gyration of a sphere whose radius is a about a diameter. In order that all points of the elements shall be at the same distance from the axis. 222 MECHANICAL APPLICATIONS. [Art. I92. we regard the sphere as generated by the surface of a cylinder whose radius is Xy and whose altitude is 2y. The surface of this cylinder is therefore A^rcxy. The differential of the volume \s> ^Ttxy dx^ and the moment of inertia is I — ^n \x^y dx = 47c ^{cp- — x) x^ dx. Putting X = asAn 0^ I = 47ra' f ' sin^ 6 cos^^ dO = ^ . Jo 15 Dividing by ~ — , the volume of the sphere, we have ^ = ^', Radii of Gyration about Parallel Axes, 193. The moment of inertia of a body about any axis exceeds its moment of inertia about a parallel axis passing through the centre of gravity^ by the product of the mass and the square of the distance between the axes. Let h be the distance between the axes. Pass a plane through the element dm perpendicular to the axes, and let r and rx be the distances of the element from the axes. Then, r, ^i, and // form a triangle ; let d be the angle at the axis passing through the centre of gravity, then ,2 _ r{ + U^ — 2ri/f cos B (i) § XV.] RADII OF GYRATION ABOUT PARALLEL AXES. 223 The moment of inertia is therefore rl dm + l^m — 2h\r^ cos d dm . . . (2j T^dm = Now Ti and 6 are the polar coordinates of dm^ in the plane which is passed through the element; hence the last integral in equation (2) is equivalent to -2/l\ X dm. But X dm is the statical moment of the body about the axis passing through the centre of gravity. Now from the defini- tion of the centre of gravity, this moment is zero ; hence^ equation (2) reduces to 7^ dm = r^ dm + Ihn ...... (3 Introducing the radii of gyration, we have also J^ = ki^B (4) 194. As an application of this result, we shall now find the moment of inertia of a cone whose height is h, and the radius of whose base is a, about an axis passing through its vertex perpendicular to its geometrical axis. Taking the origin at the vertex of the cone, the axis of x coincident with the geo- metrical axis, and a circle perpendicular to this axis as the generating element, we have for the area of this element ny^^ and for its radius of gyration about a diameter parallel to the given axis, — . 4 224 MECHANICAL APPLICATIONS. [Art. 1 94. The distance between these axes being x, the proposition proved in the preceding article gives an expression for the radius of gyration of the element about the given axis ; viz., x^ + — . Replacing r^, in the general expression for / (Art. 4 188), by this expression, and substituting for dm the differen- tial ny^ dx^ we have I = n\{x^^t^fdx, in which y — ax ' li ' Therefore 1 = nd' If '!'(■ Jo \ (' and since V- ^"'"'^ , ^=.A(,. + 4,.). To find the square of the radius of gyration about a parallel axis through the centre of gravity, we have To find the moment of inertia of a right cone about its geometrical axis we employ the same generating element as before ; but in this case the square of the radius of gyration is Hence 2 y "l\'''"%i'""- § XV.] RADII OF GYRATION ABOUT PARALLEL AXES. 22$ therefore 1= , whence ^=^^—. lO lO Polar Moments of Inertia. 195. In the case of a plane area, when the axis of rotation passes through the origin, we have r^ = ^ -\- ^, hence r^ dm = (jv^ + j^) dm, therefore /= \:t^ dm + ly^dm; that is, tke sum of the moments of inertia of a plane area about two axes in its own plane at right angles "to each other is equal to the moment of inertia about an axis through the origin perpendicu- lar to the plane. / in the above equation is called the polar mome7tt of inertia. In the case of the circle, since the moment is the same about every diameter, the polar moment is twice the moment about a diameter ; that is, denoting the former by // and the latter by /«, we have See Art. 191, Examples XV. I. Find the radius of gyration of a circular arc (2^) about a radius passing through its vertex. 226 MECHANICAL APPLICATIONS. [Ex. XV. Solution : — Taking the origin at the centre, and the axis of x bisecting the arc, ' and denoting hy 2a the angle subtended by 2s, we have mJk' = [' / ds = a' f" sin' B dQ. ^,^.V^_sin^\ 2 \ 2a J m = 2aa 2. Find the radius of gyration of the same arc about the axis of y^ and thence about a perpendicular axis through the centre of the circle. k = a. 3. Find the radius of gyration of the same arc about an axis through its vertex perpendicular to the plane of the circle. See Ex. XIV., 14, and denote by c the subtending chord. >e^^a\.-^-). 4. Find the moment of inertia of the chord of a circular arc, in terms of the diameter parallel to it, and its angular distance from this diameter. 73 See Arts. 189 and 193. / = — (3 cos a — cos ^a) . 24 5. Find the radius of gyration of an ellipse about an axis through its centre perpendicular to its plane. Eind the radius of gyration about the major axis and about the minor axisy and apply Art, 195. k' = i{a' + b'), 6. Find the radius of gyration of an isosceles triangle about a per pendicular let fall from its vertex upon the base (2b). 6- § XV.] EXAMPLES. 227 7. Find the radius of gyration about the axis of the curve, of the area enclosed by the two loops of the lemniscata r^ — a' cos 29. ^• = -3(3^^-8). «. Find the radius of gyration of a right triangle, whose sides are a and b, about an axis through its centre of gravity perpendicular to its plan^ 18 9. Find the radius of gyration of a portion of a parabola bounded by a double ordinate perpendicular to the axis, about a perpendicular to its plane passing through its vertex. 10. Find the radius of gyration of a cylinder about a perpendicular that bisects its geometrical axis, 2/ being the length of the cylinder, and a the radius of its base. 4 3 11. Find the radius of gyration of a concentric spherical shell about a tangent to the external sphere, the radii being a and b. 12. Find the radius of gyration of a paraboloid of revolution about its axis, in terms of the radius {h) of the base. 3 13. Find the moment of inertia of an eUipsoid about one of its principal axes. 15 228 MECHANICAL APPLICATIONS. [Ex. XV. 14. Find the radius of gyration of a symmetrical double convex lens about its axis, a being the radius of the circular intersection of tne two surfaces, and b the semi-axis. 15. Find the radius of gyration of the same lens about a diameter to the circle in which the spherical surfaces intersect. 2o{b' + 3«^) ' THE END. 14 DAY USE Oyi SiAiiilicS LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. i^ - ffmi — ^^ I lUULk WOV 19,1965 OCT Tm4^^ NOV .2.a-1flg 9 1?f=^'M9B8 - r-Ji J Nl l O^e fe a." 1963 ^ TTucrtrtJ ^: ,1 1/5'" hii 21-50m-6,'60 (jB1321slO)476 General Library University of California Berkeley YCI02 *