UC-NRLF $B 53D 712 ^•t:\)Vi:.^}-: Edward 3».ri^t- Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofdifferOOhallrich ELEMENTS DIFFERENTIAL AND INTEGRAL CALCULUS WITH APPLICATIONS BY WILLIAM S. HALL, E.M., C.E., M.S. PROFESSOR OF TECHNICAL MATHEMATICS IN LAFAYETTE COLLEGE THIRD EDITION REVISED NEW YORK D. VAN NOSTRAND COMPANY 1899 ^ o3 u^3 Copyright, 1897, ^ By D. van NOSTRAND COMPANY- NorfDooD ^ress : J. S. Gushing & Co. — Berwick & Smith. Norwood, Mass., U.S.A. PREFACE. This work is an introduction to the study of the Differential and Integral Calculus, and is intended for colleges and technical schools. The object has been to present the Calculus and some of its important applications simply and concisely, and yet to give as much as it is necessary to know in order to enter upon the study of those subjects which presume a knowledge of the Calculus. The book will be found to be adapted to the needs of the mathematical student, and also will enable the engineer to get that knowledge of the Calculus which is required by him in order to make practical applications of the subject. All of the formulas for differentiation are established by the method of limits. This method is preferred because it is more readily understood, and is more rigorous than the method of infinitesimals; and, moreover, it has the great advantage of being a familiar method, as the student has previously used it in Algebra and Geometry. But the differential notation is fully explained, and is employed when there is any advantage gained by so doing, particularly in the investigations of the Integral Calculus. As soon as the fundamental formulas of differentiation have been established, the corresponding inverse operations or integrations fol- low. Thus the essential unity of the two branches of the Calculus is emphasized, the whole subject is made more intelligible, and there is a saving of much space. Principal applications of the Calculus, as in Maxima and Minima, 797946 iv PREFACE. Radius of Curvature, etc., are treated at some length, while less important subjects are treated much more briefly. A large number of carefully selected examples, some original ones, and numerous practical numerical problems from mechanics and dif- ferent branches of applied mathematics are given. As there has been an increasing demand for a short course in Differential Equations, a chapter on this subject is given which it is hoped will meet a much-felt want. A table of Integrals, arranged for convenience of reference, is appended. Many American and English books, and some of the leading French and German works, have been freely consulted, and problems have been gathered from many different sources. WILLIAM S. HALL. Easton, Pa., January, 1897. CONTENTS CHAPTER I. Definitions and First Pbinciples. Art. Page 1. Constants and Variables 1 2. Functions _1 3. Increments 3 4. Limits . 4 6. Theory of Limits • . . . . . 5 6. Limiting Ratio of Increments 6 7. Derivatives . 8 8. Differentiation and the Differential Calculus 10 CHAPTER II. Differentiation of Algebraic Functions. 9. Definition 11 10. Algebraic Sum of Functions 11 11. Product of a Constant and a Function 12 12. Any Constant 12 13. Product of Two Functions . . 13 14. Product of Three or More Functions 13 15. Quotient of Two Functions , . 14 16. Constant Power of a Function. Problems 15 CHAPTER in. Differentiation of Transcendental Functions. 17. Definitions . 19 18. Base of the Natural System of Logarithms 20 19. Logarithmic Functions 20 V Vi CONTENTS. Art. Page 20. Exponential Function with Constant Base 21. Exponential Function with Variable Base. Problems 22. Circular Measure 23. Limiting Value of ^^. 24. Trigonometric Functions 26. Inverse Trigonometric Functions. Problems . 21 22 24 25 26 29 CHAPTER IV. Differentials. 26. Definition o . . . 34 27. Geometric Interpretation of -^ , . .35 dx 28. Geometric Derivation of Formulas. Problems . . c . . 36 CHAPTER V. Integration. 29. Definition 40 30. Fundamental Formulas 40 31. Elementary Rules 42 32. Constant of Integration. Problems 43 33. Integration of Trigonometric Differentials. Problems .... 48 34. Definite Integrals • . 49 36. Geometric Illustration of Definite Integration 60 36. Change of Limits. Problems . . 62 CHAPTER VL Successive Differentiation and Integration. 37. Successive Derivatives 55 38. Successive Integration. Problems . . 66 Applications in Mechanics. 39. Velocity and Acceleration of Motion . 59 40. Uniformly Accelerated Motion. Problems 60 41. Derivatives of the Product of Two Functions. Problems . . .61 CONTENTS. vii CHAPTER VII. FUNCTIONB OF TwO OR MORE VARIABLES. IMPLICIT FUNCTIONS. ChANGE OF THE Independent Variable. A KT. Page 42. Partial Differentiation 64 43. Total Differential of a Function of Two or More Independent Variables . 65 44. Total Derivative when M =/(x, y, 2), ?/ = 0(x), and 2 = 01 (a;) . . 66 45. Successive Partial Derivatives 68 46. If ?t=/(a;, y),toprovetliat-^= ^^ ...... 69 dydx dxdy 47. Implicit Functions, Problems 69 48. Integration of Functions of Two or More Variables .... 71 49. Integration of Total Differentials of the First Order. Problems . . 72 50. Change of the Independent Variable. Problems 73 CHAPTER VIII. Development of Functions. 51. Definition 78 52. Maclaurin's Tneorem. Problems . . . . . . .78 53. Taylor's Theorem , . 83 54. Demonstration of lay lor's Theorem. Problem 84 55. Rigorous Proof of- Taylor's Theorem 87 56. Remainder in Taylor's and Maclaurin's Theorems 88 57. Taylor's Theorem for Functions of Two or More Independent Variables 89 CHAPTER IX. Evaluation of Indeterminate Forms. 58. Indeterminate Forms 91 59. Functions that take the Form -. Problems -. 92 60. Functions that take the Form - 94 CO 61. Functions that take the Forms x qo and oo — oo 95 62. Functions that take the Forms 0°, 00° and li"'. Problems ... 96 63. Compound Indeterminate Forms. Problems ...... 98 CHAPTER X. Maxima and Minima of Functions. 64. Definitions and Geometric Illustration . . , , . . . 99 65. Method of Determining Maxima and Minima . . . , . . . 100 VlU CONTENTS. Art. Page 66. Conditions for Maxima and Minima by Taylor's Theorem. Problems . 101 67. Maxima and Minima of Functions of Two In 1 pendent Variables . . Ill Q8. Maxima and Minima of Functions of Tbiv e Independent Variables. Problems > 113 CHAPTER XI. Tangents, Normals and Asymptotes. 69. Equations of the Tangent and Normal 116 70. Lengths of Tangent, Normal, Subtangent and Subnormal . . . 117 71. Tangent of the Angle between the Radius Vector and the Tangent to a Plane Curve in Polar Coordinates 117 72. Derivative of an Arc 118 73. Derivative of an Arc in Polar Coordinates 119 74. Lengths of Tangent, Normal, etc., in Polar Coordinates. Problems . 120 75. Rectilinear Coordinates 122 76. Asymptotes parallel to the Axis ........ 123 77. Asymptotes determined by Expansion 124 78. Asymptotes in Polar Coordinates. Problems 124 CHAPTER XII. Direction of Curvature. Points of Inflection. Radius of Curvature. Contact. 79. Direction of Curvature 126 80. Direction of Curvature in Polar Coordinates 128 81. Points of Inflection. Problems 129 82. Curvature. Problems 130 83. Radius of Curvature 131 84. Radius of Curvature in Polar Coordinates 132 85. Contact of Different Orders . . 133 86. Radius of Osculating Circle and Coordinates of Centre . . . .134 87. Osculating Circle and Contact of the Third Order, Problems . . 136 CHAPTER XIIL EVOLUTES AND INVOLUTES. ENVELOPES. 88. Definition 138 89. Equation of the Evolute 138 90. A Normal to Any Involute is Tangent to its Evolute . . . .140 CONTENTS. ix Art. Page 91. The Difference between any Two Radii of Curvature of an Involute . 141 92. Mechanical Construction of an Involute 142 93. Envelopes of Curves 142 94. Equation of the Envelope of a Family of Curves. Problems . . 143 CHAPTER XIV. Singular Points. 95. Definitions 147 96. Multiple Points. Problems 147 97. Cusps 150 98. Conjugate Points. Stop Points. Shooting Points 151 CHAPTER XV. Integration of Rational Fractions. 99- Rational Fractions. Problems .154 CHAPTER XVI. Integration of Irrational Functions. 100. Irrational Functions 160 101. Irrational Functions containing only Monomial Surds. Problems . 160 102. Functions containing only Binomial Surds of the First Degree. Prob- lems 161 103. Functions having the Form ^^"^^^^^ 162 (a + 6a;2)2 104. Functions having the Form /(x,^^^_±_^ j da;: Problems . . . 163 ^ CX ~\~ CI ^ 105. Functions containing only Trinomial Surds of the Form Va -hbx + cx^. Problems 164 106. Binomial Differentials 166 107. Conditions of Integrability of Binomial Differentials. Problems . 166 CHAPTER XVII. Integration by Parts and by Successive Reduction. 108. Integration by Parts. Problems . 170 109. Formulas of Reduction. Problems . . . . . . , 171 X CONTENTS. CHAPTER XVin, Integration of Transcendental Functions. Integration by Series. Aet. Page 110. Introduction 177 177 178 179 182 182 111. ^Integration of the Form (/(a:)(logx)"da:. Problems . 112. Integration of the Form i x'^a'^dx. Problems 113. Integration of the Form j sin"»5 cos"i? fZ^. Problems 114. Integration of the Forms i x« Q,o&{ax)dx and ( x" s,m{ax)dx 115. Integration of the Forms i f«^sin»a:d:« and j e»* cos" a; (?x. Problems , 116. Integration of the Forms \f{x) arc sin x dx, \f{x) arc cos x dx, etc Problems Jdd Problems .... a + 6 cos ^ 118. Integration by Series. Problems 184 184 186 CHAPTER XIX. Integration as a Summation. Areas and Lengths of Plane Curves. 119. Integration as a Summation. Problems 187 120. Areas of Plane Curves in Polar Coordinates. Problems . . , 191 121. Rectification of Plane Curves referred to Rectangular Axes. Problems 192 122. Rectification of Curves in Polar Coordinates. Problems . . . 196 123. The Common Catenary 196 CHAPTER XX. Surfaces and Volumes of Solids. 124. Surfaces and Volumes of Solids of Revolution Problems . . . 199 125. Surfaces by Double Integration 202 126. Volumes by Triple Integration. Problems . . .... 203 CHAPTER XXI. I Centre of Mass. Moment of Inertia. Properties of Guldin. 127. Definitions 206 128. General Formulas for Centre of Mass. Problems . . . . . 207 129. Centre of Mass for Plane Surfaces. Problems 210 COXTENTS. xi Art. Page 130. Centre of Mass of Surfaces of Revolution. Problems .... 212 131. Centre of Mass of Solids of Revolution. Problems . . . .213 132. Moments of Inertia of Surfaces. Problems 214 133. Guldin's Theorems. Problems 215 CHAPTER XXII. Differential Equations. 134. Definition 135. Differential Equations of the First Order and Degree. Problems . 136. Homogeneous Differential Equations. Problems .... 137. The Form {ax -\- hy -\- c)dx + (a'x + biy + c')dy = 0. Problems . 138. The Linear Equation of the First Order. Problems 139. Extension of the Linear Equation. Problems .... 140. Exact Differential Equations. Problems 141. Factors Necessary to make Differential Equations Exact. Problems 142. First Order and Degree with Three Variables. Problems 143. First Order and Second Degree. Problems 144. Differential Equations of the Second Order. Problems 217 218 219 221 222 224 225 227 230 232 234 APPENDIX. Table of Integrals 238 Note A 250 DIFFEEENTIAL AND INTEGEAL CALCULUS. CHAPTER I. DEFINITIONS AND FIRST PRINCIPLES. Art. 1. Constants and Variables. The quantities employed in the Calculus belong to two classes, — constants and variables. A constant quantity is one which retains the same value through- out the same discussion. Constants are usually denoted by the first letters of the alphabet. A variable quantity is one which admits of an infinite number of values in the same discussion within limits determined by the nature of the problem. Variables are usually represented by the last letters of the alphabet. Art. 2. Functions. One variable quantity is a function of another when they are so related that for any assigned value of the latter there is a corre- sponding value of the former. Arbitrary values may be assigned to the second variable, which is then called the independent variable, while the first variable or function is called the dependent variable. For example, the area of a circle is a function of its diameter because the area depends on the length of the diameter, and the diameter whose length may be assigned at pleasure is the inde- pendent variable. B 1 2 . , ... J>IEFERENTIAL "AND INTEGRAL CALCULUS. The trigonometric functions are functions of the angle, the angle being regarded as the independent variable. ^ Expressions involving a;, such as oc^, ax^ -\- bx + c, log X, Vl — a?*, •- are functions of the independent variable x. a A quantity may be a function of two or more variables. For exam- ple, the area of a plane triangle is a function of its base and altitude ; the volume of a rectangular parallelopiped is a function of its three dimensions. The expressions, a^-Zx'if^y^ ^o'x^ + hY, a^'+y, are functions of x and y. The expressions, a'^r^j^lY^(?z% lo^ix'^-xy + z^ are functions of x, y, and z. An explicit function is one whose value is expressed directly in terms of the independent variable and constants. For example, y is an explicit function of x in the equations 2/ = - Va^ — d\ and y = 2ax-\-:»? + a?. Explicit functions are denoted by such symbols as the following : which may be read respectively: "?/ equals the / function of a;"; "2/ equals the large F function of x" \ "1/ equals the <^ function of a; " ; "?/ equals the /prime function of x." When the equation giving the relation connecting the variables is not solved with reference to y, y is an implicit function of x. For example, y is an implicit function of x in the equations DEFINITIONS AND FIRST PRINCIPLES. 3 Implicit functions are denoted by such symbols as the following : fix,y) = 0; F(x,y) = 0', ^{x,y) = 0', which may be read, " the / function of x and y equals zero " ; etc. Art. 3. Increments. If a variable receives any addition to its value, this addition is called an increment, and is usually denoted by the symbol A placed before the variable. Thus an increment received by the variable x would be denoted by ^x, and would be read " delta a;," or " increment of ic." The increment of a variable may be either positive or negative ; if it is positive the variable is increasing, and if it is negative the variable is decreasing. A negative increment is sometimes called a decrement. Art. 4. Limits. A limit of a variable is a constant value which the variable contin- ually approaches, and from which it can be made to differ by a quan- tity less than any assignable quantity, but which it cannot absolutely equal. For example, assume that a body is moving along a straight line from ^ to 5 as in Fig. 1, under the condition that in the first interval [ ' i 2 3 4 i Fig. 1. of time it shall move one-half of the entire distance, or from A to 1, and one-half of the remaining distance, or from 1 to 2, in the second interval, and so on, moving during each interval one-half of the dis- tance remaining. In this case the entire distance AB is a constant toward which the distance traversed by the moving point continually approaches as a limit but never reaches. The limit of -, as ic increases indefinitely, is zero ; as a; in this frac- tion increases the fraction decreases, and as x may be increased at pleasure, the fraction may be made to approach indefinitely near to zero. DIFFERENTIAL AND INTEGRAL CALCULUS. Let the locus of the equation ^ = - be drawn by the method of rectangular coordinates as in Fig. 2. Fig. 2. If x= 1, then y = 1; If x = 2, then y = .5-, If x = 4t, then y = .25 ; If a; = 100, then ?/ = .01 ; Or as the abscissa increases the ordinate decreases toward zero as a limit ; thus the curve continually approaches the X-axis, but never reaches it. The limit of the value of the repeating decimal 0.555..., as the number of decimal places is continually increased, is f. A variable may approach its limit in three ways : 1st. A variable may increase toward its limit, as is the case when a polygon is inscribed in a circle ; the polygon will increase toward the circle as its limit, as the number of sides is increased. 2d. A variable may decrease toward its limit, as is the case when a polygon is circumscribed about a circle; the polygon will decrease toward the circle as its limit, as the number of sides is increased. 3d. A variable may approach its limit and be sometimes greater and sometimes less than its limit. For example, take the geometrical DEFINITIONS AND FIRST PRINCIPLES. 5 progression whose first term is 1 and whose ratio is — ^, giving the series 1, — J, J? — tt> • • • ; ^^^^ ^^^ limit of the sum of the series, as the number of terms is indefinitely increased, is |; but the sum of any odd number of terms will be greater than this limit, and the sum of any even number of terms will be less. Art. 5. Theory of Limits. From the definition of a limit of a variable, it follows that the difference between the variable and its limit is a variable which has zero for its limit. Therefore, to prove that a given constant is the limit of a certain variable, it is sufficient to show that the difference between the variable and the constant has the limit zero. 1st. The fundamental proposition in the theory of limits is the ^ollowin<^: If two variables are equal and are so related that as they change they remain always equal to each other, and each a2:)proaches a limit, their limits are equal. Let X and y be the variables, and a and h their respective limits, and let a?' and y' represent the differences between the variables and their limits. Then a = x + x\ and b = y -^y'. Since x = y is always true, a —b = x' — y'. (1) In equation (1), x' — y' is equal to a constant, and x' and y' are varia* bles that approach zero as a limit. Hence x' — y' = 0, and, therefore, a — b = 0, ov a = b. The supplementary propositions are readily established. 2d. The limit of the algebraic sum of a finite number of variables is the algebraic sum of their limits. 3d. The limit of the product of two or more variables is the product of their limits. 4th. The limit of the quotient of two variables is the quotient of their limits. 6 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 6. Limiting Ratio of Increments. ^ If an increment be given to a; in 2/ =f(x), y will receive a corre- sponding increment ; required the limiting value of the ratio — ^• Aa; Taking first a particular function, for example, y — ax^. In this example, if x receives an increment represented by ^x or /i, y will take a corresponding increment represented by Ay] and the equation becomes y-\-Ay = a(x-\-hy = ax^ -\-2 axil -f- aAl Subtracting y = aa^^ Ay = 2axh-\-ah\ (1) Dividing by Ax = h, ^=2ax-i-ah. (2) Ax As 7i approaches zero, each member of this equation will approach a limit, and by Art. 5 these limits are equal ; therefore limit of ^ = 2 ax. (3) Ax ^ ^ In order to make a definite application, let a = 16 in the given equa- tion, and substitute s for y, and t for x\ then the equation becomes s = 16 fy which is approximately the equation of a freely falling body near the earth's surface, s representing the number of feet fallen, and t the time of the fall in seconds. Then the proper substitutions made in equation (3) give limit of — = 32^, which is seen to be the actual velocity at the end of t seconds. There- fore the limiting ratio of the increments of distance and time is the velocity at the end of the period. To illustrate further, let the object be to determine the increments produced in s by certain decreasing increments assigned to t, when t has some given value, as 10. DEFINITIONS AND FIRST PRINCIPLES. 7 Substituting s = 2/, ^ = a; = 10, a = 16 and At = Aa;, in (1), (2) and (3): As = 320At-\-16(M)% |^ = 320 + 16(A0, and limit of — = 320. At Let At = 0.1, then As = 32.16 and — = 321.6 ; A^ ' Let A^ = 0.01, then As = 3.2016 and — = 320.16 : A^ ' Let A^ = 0.001, then As = .320016 and — = 320.016 ; At ' Let A^ = 0.0001, then As = .03200016 and — = 320.0016 ; A^ ' And it is apparent that as A^ continually diminishes, As also As decreases, and the ratio of the increments, — , approaches 320 as its limit. Take next a geometrical example. Let the curve be the parabola whose equation is y = V2 x^ and whose locus is shown in Fig. 3. Let (x', y') be the coordinates of P, and (x' + Ax, y' + Ay) be the coordinates of any second point P'. Then y' + Ay = ^2 (x' -\- Ax). (1) Subtracting q^ y'=^2x', ^-y^^ Ay = ^2(x' + Ax)-V2^', hence " '^^ Ay^V2(x' + Ax)-V2^'^ ^ Ax Ax ^ ^ Rationalizing the numerator of (2), Ay 2 Air 2 — ^ = — — — = — — ; i :s^ Aa; Aa;[V2(x' + Aa;) + V2a;'] V2 (x' -^ Ax) -{- ^/2x' ^^ and limitof ^ = — 4= = -^ = i. ^^ ^a; 2V2a;' V2x' y\^' 8 DIFFERENTIAL AND INTEGRAL CALCULUS. From the figure, it is obvious that — ^ is the tangent of the angle Ax P^TN, and if the point P' approaches indefinitely near to P, the line PT will be a tangent to the curve at P. Therefore, the limit of -^, as Ax Fig. 3. Ax approaches zero, is the tangent of the angle which the curve makes with the X-axis, and is equal to the reciprocal of the ordinate of the point of contact. If P is at the extremity of the latus rectum, coordinates (-J, 1), then limit of ^ Ax tan 45= or the tangent to the parabola at the extremity of the latus rectum makes an angle of 45° with the X-axis, which is a well-known property of the curve. Art. 7. Derivatives. The limit of — ^ in the preceding article is called the derivative of y ^^ . dv with respect to x, and is denoted by — • Hence the definition : If y is cix a function of x, the derivative of y with respect to x is the limiting DEFINITIONS AND FIRST PRINCIPLES. 9 value of the ratio of the increment of y to the corresponding increment of ic, as the increment of x approaches zero. In general, let y=zf{x). (1) When X is given an increment Aa? or h, y takes a corresponding increment Ay, and the equation becomes y + ^y=f{x + h). (2) Subtracting (1) from (2), ^y=f{x + h)-f{x)._ (3) Dividing (3) by Ax, Ay ^ /(o; + /^ -/(a;) ^ ^^^ As Ax approaches zero, the limit of — ^ is the derivative of the dv ^^ function, and is represented by -^« dx Therefore ^^ = SM = limit of fi^ + K)~m _ dx dx Ax The derivative is often called the differential coefficient, and the symbol /'(x) is frequently used instead of --^• The term "derivative " is fully as significant as differential coefficient, and is certainly to be preferred when the method of limits is used. The word "deriva- tive " will generally he used in this book. Derived function is another name which is sometimes adopted instead of the word "derivative." It must be carefully noticed that A and d are not factors, but symbols of operations. The general method of finding the derivative of any function of x is as follows : Two values of the independent variable, as x and x + Ax, are taken, and the corresponding values of the given function are found; the difference between these two values of y is the increment of the function corresponding to the increment Ax given to x. The limit of the ratio of these two increments, as Ax approaches zero, will be the derivative of the function. According to this method, general rules or formulas are obtained for forming the derivatives of the different kinds of functions. 10 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 8. Differentiation and the Differential Calculus.* The operation of finding the derivative of a function is called differentiation. The object of the Differential Calculus is to determine the deriva- tives of functions, and to investigate their properties and applications. f J PROBLEMS. 1. In the equation 2/ = £c^ — 3ic + 5, what is the increment received by y if an increment of 1 is given to x, when ic = 3 ? Ans. 4. 2. In the equation y — mx -\- n, what is the ratio of the increment of the ordinate to the increment of the abscissa ? A^is. m. 3. In the equation y^ = ^(ISx — x^), what are the increments received by y corresponding to an increment of 1 given to the abscissa, starting from x = S? Ans. — 2 V5 ± fVli. * The Differential and Integral Calculus originated in the seventeenth century. Newton was the first discoverer of the new analysis, but to Leibnitz belongs the credit of priority of publication and the invention of a notation much superior to Newton's, and which has entirely superseded it. Leibnitz first published his new method in 1684. Newton called his method the method of fluxions. According to him, all quan- tities are supposed to be generated by continuous motion, as a line by a moving point. Fluxions are the relative rates with which functions and the variables on which they depend are increasing at any instant. Leibnitz considered all quantities to be made up of indefinitely small parts or infinitesimals ; a surface being composed of indefinitely small parallelograms, and a volume of indefinitely small parallel opipeds. The nomenclature and notation of the Calculus now in common use were original with Leibnitz, and introduced by him. According to Newton, the fluxion of x would be denoted by x, while by Leibnitz dx the corresponding derivative is — dt > -.u 11 ^ f^ ;.% ^^ 4^1" CHAPTER II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. Art. 9. Defixition. An algebraic function is one in which the only indicated operations are addition, subtraction, multiplication, division, and involution and evolution with constant exponents.* In this chapter only functions of a single independent variable x will be treated, and throughout the chapter u, v and w will be regarded as functions of x. .^ Art. 10. Algebraic Sum of Any Number of Functions. If y be taken to represent the algebraic sum of three functions of Xj the equation may be written y = u-\-v — w. (1) If an increment Ax is given to x, the variables y, u, v and Wy which are functions of x, will take the corresponding increments Ay, Ati, Av and Aw, respectively ; then (1) becomes y -\- Ay = (u -\- Au) -{- (v + Av) — (w + Aw). (2) Subtracting (1) from (2), Ay = Au -{- Av — Aw. (3) Dividing by Ac., Ay ^ A« ^ A^ _ A«,_ Ax Ax Ax Ax When Ax approaches zero, limit of ^ = ^, limit of ^ = ^, etc., by Art. 7. Ax dx Ax dx * In this definition of an algebraic function it is understood that the operations are not repeated an infinite number of times. 11 12 DIFFERENTIAL AND INTEGRAL CALCULUS. Therefore ^ = ^ + ^ _ ^, by Art. 5, 1st and 2d ; ax ax ax ax (fB d(u-\-v — w) _dudv dw ^^ i dx dx dx dx . VJ-^ If- the algebraic sum of four or more variables be given, the deriva- tive would be found similarly. I. Hence, the derivative of the algebraic sum of any number of func- tions of X is equal to the algebraic sum of their derivatives. Art. 11. Product of a Constant and a Function. Let a represent any constant, then the product of a constant and a function of x may be written y = av. (1) Let V and y take the increments Av and ^y corresponding to the increment Aa; given to x, then y-\-Ay = a(v + Av). (2) Subtracting (1) from (2), Ay = aAv. Dividing by Aa;, _ ^=a — • (3) When Aa; approaches zero, by Art. 5, 1st, limit^ = limitfa^^; Aa; V ^^J therefore ^ = a ^, by Art. 7. dx dx If v = x, Av = Aa; and -^ = a. dx 6 II. Hence, the derivative of the product of a constant and a f motion of X is the product of the constant and the derivative of the variable. Art. 12. Any Constant. As the value of a constant remains unchanged in any one discus- sion, the constant receives no increment, or, in other words, the incre- ment of the constant is zero. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 13 Let a represent any constant ; then Aa = and ~ = 0. Ax Therefore, when Ax approaches zero, dx III. Hence, the derivative of a constant is zero. o Art. 13. Product of Two Functions. Let the product of two functions of x be represented hy y = uv. When X is given an increment, the variables v, u and y receive corre- sponding increments, and the equation becomes y -^ Ay= (u + Au) (v + Av) = uv -\- uAv -\- vAu + Au Av, (1) Hence Ay=(v-{- Av) Au + uAv ; "^ (2) and ^ = (v + ^v)^ + u^. (3) Ax Aic Ace When Ace approaches zero, limit ^ = !^, limit «^ = «*!, Aa; dx Ax dx limit (v + Av) = v, and limit ^ = ^. Ace dx Therefore, by Art. 5, dy _ d (uv) _ diu d/v /^TvJ dx dx dx dx v j/ IV. Hence, the derivative of the product of two functions of x is the sum of the products of each function by the derivative of the other. Art. 14. Product of Three or More Functions. Let the product of three functions of x be represented hj y = uvw. The product of two of the functions, as uv, may be taken equal to z ; then, by the preceding article, 14 DIFFERENTIAL AND INTEGRAL CALCULUS. d(uvw) d(zw) dz , dw ., , dx dx dx dx * ^ Z r ac \ dx dx) dx * du , dv , dw = wv \-ivu ^uv • dx dx dx G This process may be extended to the differentiation of the product of any number of functions. V. Hence, the derivative of the product of any number of functions of X is equal to the sum of the products of the derivative of each 'into the product of all the others. Art. 15. Quotient of Two Fuxctioxs. Let the quotient of two functions of x be represented by 2/ = — Then vy = u; and, by V., dx dx dx Therefore du dv dy dx dx dx * dx dx /! \ = — ^ — (>i.) VI. Hence, the derivative of a fraction is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multi- plied by the derivative of the denominator, divided by the square of the denominator. du CoR. 1. If the numerator is constant, — = 0, by III., and VI. dx becomes „. dv u — dy_ dx dx~ v^ DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 15 Hence, the derivative of a fra ction with a constant numerator is ( negative and equal to the numerator multiplied by the" derivative of j the denominator, divided by the square of the denominator. CoR. 2. If the denominator is constant, — = 0, by III., and VI. dx becomes dy ^ dx v^ du du dx dx which is the same result that would be obtained by II. Art. 16. Constant Power of a Function. Case 1. When the exponent is a positive integer. Let -y be a function of x, and n its exponent ; then y = V", 2/ + A?/ = (v + Av)", and Ay = (iJ H- A?;)" — v". Expanding (y -\- Av)", by the Binomial Theorem, and dividing by Aa, Ay Ax -^v"-2(Av) ... +{^vY ' Li A'y Aa; When Ao; approaches zero, Av approaches zero also; hence d\i „_i dv dx dx Case 2. When the exponent is a positive fraction, m Let y = v''^ (t/ then ?/** = v"". As m and n are positive integers, by Case 1, dx dx therefore dy_m v"'^^ dv _m i;"*'^ d^ dx n v""^ dx n m-- dx ^m^-^dv^ n dx VIL) ^- 5»w *• r<.( «i «.r •1-1 - M vVir-.;. ^-. ivr,.]*" Ui 10 DIFFERENTIAL AND INTEGRAL CALCULUS. Case 3. When the exponent is negative and either integral or fractional, as — n. Let y = v~", then y = — Differentiating by Art. 15, Cor. 1, da; 'y'" dx • ^ VIT. The derivative of a constant power of a function of x is equal to the product of the exponent, the function ivith its exponent diminished by unity, and the derivative of the function. Radical expressions may be differentiated according to this rule, the quantities being first transformed into equivalent expressions with fractional exponents. The radical of the second order is the one that occurs most fre- quently. It is differentiated as follows : Let y = Vv = v2. Hence, the derivative of the square root of a function of x is equal to the derivative of the function divided by twice the square root of the function. PROBLEMS. The formulas established in this chapter are sufficient for the differentiation of all algebraic functions of a single variable. Differentiate the following functions : 1. y z=z a + bx ■{■ :x?. |=d> + ^-^^) ^^^IM + ll^, byL dx dx dx ^ DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 17 ^ = 0, by III. ; ^-^^ = 6, by 11. ; ^^ = 3a^, by VII. dx dx dx Therefore ^=5 + 3ar^. dx 2. y= (a+x)(b-{-2af). ^=(b + 20^ %t^) + (a + a;)^(^±2^, by IV., dx dx dx = (b-\-2x^-^4:(a-\-x)x = b-{-6x- + Aax. 3. V = • ^ (a + x'f Applying VI., and VII., dy_ {a + a^yx 12o^ - 4:X^ x S(a -\- x'Y x (2x) dx (a + i^y ^ 12x^(a-x^ (a + ary 4. y = x(l + x'){l-^o^). ^^=(1 -\-ar^(l ^ :^) + x(l + af)£(l +a^ + x(l -\-x^)£(l +x') . = (1 + x2)(l + ar3)+ a^(l + a^)(2a')+ a;(l + a:2)(3a?2) \-\-x dy _ 1 — 2x — a^ 5. y = l+x" dx (l-^-x'f r- dy a ■ 7. . ?/ = (a+x)'"(6+a;)". ^ = (a + a;)'"-^(6 + x)'*-i[m(6 +«?) + w(a + a;)]. 9. 2/=-^^ ^ = _ JL.. dy _ _ g + 3x (?a:~ 2VaT^' 10. yz= (a — x) Va + a:. 18 DIFFERENTIAL AND INTEGRAL CALCULUS. / 2 . o^ /-2 -2 - cly_ a'-{-a^a:^-4:X* 11. y = x(a^-^x-)Va^ — af. -^= , if \ ^ ) clx -yja'-y? 12. 2/ = dy_ 1 •Vl-a^ ^^^ ■yJl-x^ + 2x{X-x') 13. y^ Vl + a^ + Vl-g; . 'dy^ l-f-Vi-a;^ dx 3a^ + 2 dy^ 2 16. ^^vTTZ±V^^. ^ = _2^^ 14. y=(l-Zx'-\-Q>x%l + x')\ ^ = 60a^(l + £c2)2^ O ^ I o 15. y = 3 (aj2 _^ 1)1(4 a^ _ 3). ^ = 56 cc3(a^ + l)i clx 2a;^-l ^ = J_±i^. a; Vr+¥2 da; ^2(-l _|_ ^,2)1 1 Q ., — V(^ + c^)^ ^ _ (a? — 2 g) Va; + g ly. ?/ — • — g • Va; — g "-'^ (a;_ay2 20 ^^ ^ ^ (Zy^ m(6 + a;)+n(a + a?) . (g + a;)*" (6 4- a;)" da: (g + xY-^\b + a;)"+i 21. y=( E Y cZy_ ny 22 . = ^/-iEZ. dy_ 2a:(2-a^ \(l + a;2)« da; (X-x^)l(l^^)^ CHAPTER III. DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. Art. 17. Definitions. All functions that are not algebraic are called transcendental. Tran- scendental functions are divided into four classes : 1st. Logarithmic functions ; those in which a logarithm of a variar ble is involved. 2d. Exponential functions ; those in which a variable enters as an exponent. 3d. Trigonometric functions; those involving sines, cosines, tan- gents, etc., in which the arc is the independent variable. 4th. Inverse trigonometric functions; those derived from trigono- metric functions, by taking the arc as the dependent variable. Thus, from the trigonometric function, y = sin x, is obtained the inverse function, x = arc sin y, which is read, " x equals the arc whose sine is 2/." The inverse trigonometric functions are also called circular func- tions and anti-trigonometric functions. The inverse trigonometric functions are often expressed differently, as shown in the following identities : arc sin?/ = sin^^ y ; arc tan y = tan "^2/ 5 ^^^ cosec y = cosec"^?/. This second notation, employed to express inverse trigonometric func- tions, was suggested by the use of negative exponents in algebra, but the student is cautioned against the error of regarding sin~^2/ ^^ equiva- lent to -: Another application of this notation for inverse func- smy tions is seen in an anti-logarithm; ii y = logo;, then x = log'^y. 19 20 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 18. Base of the Natukal System of Logarithms. The base of the natural system of logarithms is the limit of fl;-f-J as X approaches infinity. By^-the Binomial Theorem, = 1-1-1 4 U .^^ i-^ L -4- .... ^ ^ 1.2 ^ 1.2.3 ^ When X increases indefinitely, , This limit is usually denoted by e. Therefore e = H- 1 + r^^ + —i-^ + -• By summing this series the value of e is found to be 2.7182818+, which is the base of the natural system of logarithms. Art. 19. Logarithmic Functions. Throughout this chapter, v and u will always be regarded as func- tions of a single independent variable x. Let the base of the system of logarithms be a ; then let y — log, v ; hence y + A^ = log, {v + Av), A2/ = log, {v + Av) — log, V V \ V J DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 21 ^=^iogji+— y°- Av and — ^ = — ^» ■ . Ax V \ V J Now as Ax. approaches zero, Av approaches zero; and therefore — approaches zero and — approaches infinity. If — be substituted for x in the preceding article, the limit of (-?) IS seen to be e. dv Therefore ^=log,e^. YIII. clx V Log„e is the modulus of the system in which the logarithm is taken and may be denoted by M. VIII. Hence, the derivative of the logarithm of a function of x is equal to the modidus, multiplied by the derivative of the function, divided by the function. Hereafter, when no base is specified, it will be understood that natural logarithms are used ; then Jbr=log«e = log,e = l, aud VIII. becomes ^ = — -^ VIIL a. dx dxv ■ Art. 20. The Exponential Function with a Constant Base. Let the exponential function with a constant base be y = a\ Taking the logarithm of each member, log y=vlog a. Differentiating by VIII., dy ,^dx , dv M — = loga3-; 22 DIFFERENTIAL AND INTEGRAL CALCULUS. IX. dx Ajid when natural logarithms are used, therefore dy^a^gaclu_ dx M dx •: f? = „.logaf!. IX.a. dx dx If a = e in IX. a, since log^e = 1, iiL^e^. IX. 6. dx dx li v = xm IX. a, ^^ = a^ log a. (1) Ifa = ein(l), ^ = e^ IX. c. dx IX. Hence, ^^e derivative of an exponential function ivith a constant l)ase is equal to the function multiplied by the logarithm of the base and by the derivative of the exponent, divided by the inodidus. Art. 21. The Expoxential Function with a Variable Base. Let the exponential function with a variable base be y = u\ Then log y = v\o%u] dy du -^ V — and by VIII., M — = M— h log u —' y u ax mi, £ dy „_, du , u" log u dv v Therefore -f- = vu"^ —- H r-^ — - • X. dx dx M dx X. Hence, the derivative of an exponential function with a variable base is equal to the sum of tioo derivatives ; the first being obtained as though the base ivere variable and the exponent constant, and the second as though the base were constant and the exponent variable. DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 23 PROBLEMS. 1. y=ia:^\ogX. 2. ?/ = log (2 a; + o^. a-\-x 3. 2/ = log a — x 4. 2, = logV|g- 5. 2/ = log(^ + Vl H-ar^. 6. 2/ = loga^. 7. 2/ = log^^' 8. y= log (logic). , -Vx^ -\-l—x 9. ^ = log-— ^= VaH H- 1 + aj 10. 2/ = l0g--=^r^^— ==:• Vl 4- »— Vl —a; 11. 2/ = log(VH-a)'+Vl-ar). 12. 2/ = a**- 13. ?/ = «''• ^ = 2a''.loga.a?. 14. ?/ = e*(a; — 1). da; : a; (2 log a; 4-1). c/2/_ 24-3ar^ ca- 2a; + a^ cly_ 2a dx a'-x" dy_ 1 dx 1-ar^ dx 1 Vl4-ar^ dy_ dx _2^ x ^V- _21oga; dx x dy 1 dx X log X dy_ 2 dx Va;2 4-1 dy 1 dx x^l-^ dy _ 1 1 dx- ^ X VI - a;^ dy_ dx = a'^'e^loga. dx dx 15. 2/ = 2eV5(a;'^-3a;4-6a;^-6). ^ = a;eVi. Ota; 24 DIFFERENTIAL AND INTEGRAL CALCULUS. ' e^ + 1 K. V- '' ^ 1+a; 18. •- 1 2/ = af. 19. y = af a*. 20. i/ = af. 21. 2/ = a;^^^''. ■ 22. y = yf. dx {e + lf dy xe dx (! + »;)» dy_ .af(l- logic) da; = a^^a;'*-^ (n 4- a; log a), da; = af (loc ^a; + l). dy_ dx = log a.-^ , ^logx-l ^^ = af" Aog a; + log^ x + ^V. 23. y = e'[xr—nyf-'^+n{n-l)x^-^ ]. ^ = e=»a;^ cta; Art. 22. Circular Measure.* In higher mathematics, angles are not measured by the ordinary degree or gradual system, but in terms of another unit. The circular measure of an arc of a circle is the ratio of the length of the arc to the length of its radius ; and it is evident that this ratio does not vary with the radius. Thus the value of an arc of 360° in circular measure is ^= 27r, of 180° is IT, of 90° is J, and of 1° is ~ r 2 180 The angle at the centre of a circle subtended by an arc equal to the radius is the radian or circular unit. Let X denote the number of degrees in an angle, and z the number of radians in the same angle; then since there are tt radians in two right angles, X _ z 180 ~ 7 Therefore z = -^ x, 180 * Hall's Mensuratioiu §§ 9, 10, 11, and 12. DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 25 and 180 x= z. IT Hence, to reduce from gradual to circular measure, the number of degrees in the angle is multiplied by 180 and to reduce from circular to gradual measure, the circular value is multiplied by 180 Art. 23. Limiting Value of SIX ^ Let the small angle AOB in Fig. 4 be represented by $, and the radius OA by a; and let BC, AB and AD be sin^, chords and tand, respectively. C A Fig. 4. The area of the triangle AOB = \a^ sin^; The area of the sector AOB = \a^d-, The area of the triangle AOD = \o? tan^; and these areas are obviously in an ascending order of magnitude ; hence tan ^ > ^ > sin ^, tan^ or 6 Thus - — - lies between . sm 6 sm 6 sin 6 sin 6 tan^ >1. and 1; but when approaches zero, e -1^ or approaches 1 ; hence, as 6 diminishes indefinitely, -^ — - sind cos^ FF > J j^ g.^^ approaches the limit unity. 26 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 24. Trigonometric Functions. 1. Differentiation of sin v. Lfet y = sin v, then y-}-Ay = sin (v + Av) ; therefore Ay = sin (v -f Ai?) — sin v. By Trigonometry, sin.4 - sin5= 2 cos|-(^ + 5) sin^ (.4 - B). Substituting v -^ Av = A, and v = B, in this formula, A3/ = 2cosfv+— jsin — ; sm hence — = cos ( v -{-- — Ax \ 2 J ^ Ax 2 When Aflj approaches zero, Av approaches zero, and by Art. 23, . Av sm — limit is unity. Ay ^ 2 Therefore ^=cos'y^. XI. ax ax XL Hence, the derivative of sin v is equal to cos v multiplied by — • dx 2. Differentiation of cos v. Let y = cosVf' then Ay = cos (v + Av) — cos v. By Trigonometry, cos ^ - cos 5 = - 2 sin ^(A-{-B) sin ^(A-B). DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 27 Substituting v -\- Av = A, and v = B, A2/ = -2sin(. + f)sinf; sin^ hence ^^-smfv^^] ^^. Ax \ 2 J Av Ax 2 Therefore ^ = _ sin v — . XII. dx dx XII. Hence, the derivative of cosv is negative^ and equal to sinv multiplied by — • dx Let y = tan v 3. Differentiation of tan v. sinv cosv cos V — (sin v) — sin V — (cos v) o dv , • o dv cos^v hsm^v- dx\QosvJ cos'^v dv , • 2 dv cZa; die cos^v = sec'v—' XIIL da; XIII. Hence, the derivative of tanv is equal to se(^v multiplied 4. Differentiation of cotan v. •r . . cosv Let 2/ = cotan v = smv . f . dv\ f dv\ smvf — sm V-— — cosv (cosv — ) By VI. ^ / cosv\ ^ V ^ V ^^J *' dx\mivj sin^v = -cosec2v— . XIV. dx XIV. Hence, the derivative of cotan v is negative^ and equal to cosec^v multiplied by dx By Art. 15, Cor. 1, 28 DIFFERENTIAL AND INTEGRAL CALCULUS. 5. Differentiation of sec v. , Let y = secv = • ♦. cos V d / s • dv — (cos-y) sinv — dy^d r 1 \^ dx^ ^ _ dx dx dx\GosvJ cos^v "~ cos^v = sec V tan v — • XV. dx XV. Hence, the derivative of sec v is equal to sec v, multiplied by dv tanv, into — • dx 6. Differentiation of cosec v. Let y = cosec v By Art. 15, Cor. 1, sm V dy^ df 1 \ ) dv cos?; — dx dx dx\smvj sm^y = — cosec V cotan v—- XVI. dx XVI. Hence, the derivative of cosec v is negative^ and equal to cosec Vj dv multiplied by cotan v, into — dx 7. Differentiation of vers v. Let y = vers v = l — cos v. Then ^ = — (1 - cos v) = sin v ^. XVII. dx dx dx XVII. Hence, the denvative of vers v is equal to sin v into — • ' dx 8. Differentiation of covers v. Let y = covers v = l — sin v. Then ^ = — (1 - sin v) = - cos v — • XVIII. dx dx dx DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 29 XVIII. Hence, the derivative of covers v is negative^ and equal to . . dv cos V into — dx Art. 25. Inverse Trigonometric Functions. It should be remembered that there are two ways of indicating the inverse trigonometric functions. The functions arc sin x, arc cos x, arc tan ic, etc., are often written as follows: sin~^ic, cos~^a;, tan ""■'a;, etc., respectively. 1. Differentiation oi y =■ arc sin v. Then v = sin y. By XL, T = "^'yT-^ dx dx hence dy _ 1 dv __ 1 dv dx cos y dx Vl — sin^w ^^ dv Therefore ^ (arc sin ^)^ dx ^ . dx VI —v^ 2. Differentiation of y = arc cos v. Then v = cos y. ByXIL, ^ = -smy^', ^ ' dx ^ dx' hence dy _ 1 dv 1 dv dx sin y dx Vl — cos^y ^^' dv XIX. Therefore ^(^rccos.;)^ dx__^ , ^x. dx Vl-'y^ 3. Differentiation of y = arc tan v. Then v = tan?/. hence ^ = _i_^=- 1 ^. c?a; sec^ ?/ (^x 1 + tan^ y dx 30 DIFFERENTIAL AND INTEGRAL CALCULUS. dv Therefore d (arc tan v) dx dx ~l+v''' 4. Differentiation oi y — arc cot v. Then V =coty. By XIV., dv 9 dv — = -cosec2 2/-^; dx "^ dx lence dy _ 1 dv _ 1 dv dx cosec^ ydx 1 + cotan^ y dx dv Therefore d (arc cot v) dx XXI. - , , o- XXIL dx 1 -f v^ 5. Differentiation ofy = arc sec v. Then v = sec y. By XV., ^= sec ^ tan 2/^; da; da; hence dy^ 1 di;^ 1 dv^ dx secytB,nydx sec?/ Vsec^y - l^^a; Therefore djavcseov) ^ ^ . XXIII. da; i; V-y*^ - 1 6. Differentiation ot y = arc cosec v. Then v = cosec y. By XVL, ^= - cosecycoty^; dx dx hence ^ = -_-J_^ = _^____^. da; cosec 2/ cot 2/ da; cosec ?/ Vcosec^y - 1 ^« dv Therefore cZ (arc cosec v) ^ _ da; ^ ^-j^j^ da? V Vv^ — 1 DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 31 7. Differentiation of y = arc vers v. Then v = vers y. By XVII., ice ^ = dx dx dx 1 dv 1 dv dx Therefore Vl — cos^i/ ^^' V2 vers y — veis^y^^ do d (arc vers v) _ dx ^ XXV. dx V2 V — v^ 8. Differentiation oi y = arc covers v. Then v = covers y. ByXVIIL, *' = -cos2/^=-Vl-sin'2,^; dx dx dx hence ^ = ^ ^= ^ — ^. da; Vl — sin^?/ ^^ V2 covers y — covers^ ?/ ^^^ Therefore d (arc covers 7>) ^ -Jf XXVL da; \/2 V — v^ PROBLEMS. 1 . 2/ = sin na;. -^=n cos nx\ da; 2. ?/ = sin*'a;. -^= nsin^^^ajcosa;. ^ da; 3. 2/ = cot2(a;»). ^ = - 6 a;^ cot (a.-^) cosec^ (a;^) 4. 2/ = logCsin^a;). -^=2 cot a;. da; 6. 2/ = sin 2 a; cos a;. -^ = 2 cos 2 a; cos a;— sin 2 a; sin a;. dx 6. 2/ = e*cosa;. -^ = e='(cosa; — sina;). da; 32 DIFFERExXTIAL AND INTEGRAL CALCULUS. 7. j/=e««^^sina;. ^^= e'^'' (cosx - sin^x). clx ^ «. y = smlogx. ^ = icos(loga;). clx X ^ ° ^ 9. i/= (cosa;)"°^ -^= (cosa;)"°*[cosa;logcosa; - sinajtana;]. 10. y = logt3inx. ^ 11. y = logJ\±^. ^ 1 — Sin a; 12. y = logsecx. 13. 2/ = arc sin-, a 14. 2/ = arc tan?. a 1 — a^ 15. 2/ = arc sin 16. ?/ = arcsin(3a; — 4a:^. 17. 2/ = arc sec 2a?. 2x 18. ?/ = arctan 1-ar^ 19. y = x Va- — a;^ + a^ arc sin -• 20. 2/ = 6^ dx sin2aj dx 1 cos a; dx = tana;. dy_ 1 dx Va^-ar' dy_ a dx a^^y? dy_ 2 dx 1-hx^ dy^ 3 : -. dx VI -x' d^_ 1 dx xV^x^-1 dy^ dx dy_ dx 2 l+a;^ :2Va=^-a;^. dy_ garctanx clx l+r' 21. 2, = V^-r^ + aarosm-- _ = (^__j. 22. 2, = arc tan ^^:^^. ^ = 1. VI + COS a; dx 2 DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 23. 3/ = arccot« + logJ^. dy_2a^ X ^x -\- a „ . X (arc sin ic) , , /^ -s 24. y= ^ ^ + log VI — ar. 25. 2/ = ilog(^^' + -^arctan2.^. 26. y^^"^(^sina^-cosa;)^ ^ = e-siiia.'. a^ + 1 dx a;V5 di/ 1 27. 2/ = arc sec -^ — dx x'- a* dy_ arc sin x dx (1- -.^)t dy _ 1 28. 2/ = log\/—i_^ + ^ arc tana?. ^1 — X 2Var^ + a;-l ^^ x^x^ + x-1 dy^ 1 daj 1 — a;* •: CHAPTER IV. DIFFERENTIALS. Art. 26. Introduction. The formulas for differentiation given in the preceding chapters have been established by the method of limits. In this chapter another method of treatment will be presented which is called the method of infinitesimals. According to this second method, the independent varia- ble is supposed to change by the continued addition of an infinitely small constant increment. This increment is called the differential of the variable, and the corresponding increment of the function is called the differential of the function. The differential of a variable may then be defined as the difference between two consecutive values of the variable. Hitherto, the symbol -^ has been regarded as a whole, but dx here it is defined as the ratio of the differential of the function to the differential of the independent variable, and is regarded as a fraction. The phraseology and notation of the two methods are different, but they give identical results. To illustrate : Let y=^, thenbyVIL, ^ = 5x\ dx If differentials are used, the equation becomes dy = 5 x*dx, which would be read, "The differential of y is equal to 5x^ times the differential of x." In general, let ^ = /'(ic), dx then dy = f'(x) dx. 34 DIFFERENTIALS. 35 Now the reason for sometimes calling the derivative the differential coefficient is apparent, as it is seen to be the coefficient of dx in the differential of f(x). If each member of each of the formulas, I.-XXVI., be multiplied by dx, a corresponding set of formulas will be obtained for the differ- entials of functions.* Art. 27. Geometric Interpretation of -^• dx The two methods may be compared geometrically. In Fig. 5, let AB represent any plane curve whose equation will show the relation between the coordinates of any point of the curve ; then the ordinate y may be expressed as a function of x, giving for the equation y = f(.^)- 1. By the method of limits. Let (x, y) be the coordinates of any point P of the curve and {x + Aaj, y + A?/) the coordinates of any second point P% and i/^ the Fig. 5. angle which the tangent to the curve at P makes with the X-axis. If 6 be the angle which the secant through P and P' makes with the X-axis, and PM be drawn parallel to OX, * Lagrange in Mecanique Annlytique says : " When we have properly con- ceived the spirit of the infinitesimal method, and are convinced of the exactness of 2§ DIFFERENTIAL AND INTEGRAL CALCULUS. tan^ = ^=^. PM Ax ^ Now, suppose that P' approaches P, or, in other words, that Ax decreases toward zero ; evidently approaches ip, and tan approaches tan j/^ as its limit. By definition, limit ^ = ^. Ax dx Therefore, by Art. 5, tan ^ = -^. •^ ' ^ dx 2. By the method of infinitesimals. Let P and P' represent consecutive points on the curve, then PN and P'N' are consecutive ordinates. The part of the curve, PF, called an element of the curve, is regarded as a straight line, and when pro- longed it forms the tangent to the curve at the point P. PM is drawn parallel to OX; then PM=NN'=dx, and MP' = dy. And as ., ^ = ^, ^ PM dx Hence, the derivative of the ordinate at any point of a plane curve with respect to the abscissa is equal to the tangent of the angle which the tangent to the curve at that point makes with the X-axis.* Art. 28. Geometric Derivation of the Formulas for the Differentiation of the Trigonometric Functions. In Fig. 6, let AP represent a circular arc x, with radius = 1, and PP' = dx an infinitely small increment given to x. PS is drawn par- allel to OA, and PN and P'M are consecutive ordinates. the results by the geometrical method of prime or ultimate ratios, or by the- ana- lytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of abridging and simplifying our demonstrations." * The student will be much benefited by plotting curves whose equations are of the form y =f(x), and interpreting the derivative obtained from each equation. DIFFERENTIALS. 37 PN= sinaj ; P'M= sin (x + dx) ; therefore P'S = d sin a;. 0N= cos ic ; 0M= cos (ic -|- da;) ; therefore NM= — d cos x. The triangle PP'^ is a right triangle, and Z PP'S = Z POiV. Hence, cZ sin a; = P'S = PP' cos PP'S = cos a; da;, and d cosa; = - MN=- PP' sin PP'^S = — sin X dx. XII. AT = tana;, and ^T' = tan (x -\- dx) ; hence TT' = d tan x, and CT' = cZ sec x. BD = cot X, and JB^ = cot (x -f da;) ; hence HD = — d cot a;, and HE = — d cosec a;. From the triangles CTT' and ITDJ^;, similar to NOP, the differentials of the remaining trigonometric functions may be obtained. It will be noticed in this article, that the differential of a function is negative when the function decreases as the independent variable increases. PROBLEMS. 1. If the side of an equilateral triangle increases uniformly at the rate of 2 inches per second, at what rate does the altitude increase ? Let a; = a side of the triangle, and y its altitude ; then 2/^ = f x\ Differentiating, and solving for dy, gives dy = ■—- dx, which shows that if an infinitely small increment is given to x, the corresponding incre- ment of y is — - times as great ; that is, the altitude increases ^~ times 2i 2 38 DIFFERENTIAL AND INTEGRAL CALCULUS. as fast as the side. When x is increasing at the rate of 2 inches per second, y is increasing at the rate of ^- times 2, or V 3 inches per second. # Remark. In these examples the differentials are regarded as rates. The rate of change of a variable at a given instant may be here defined as the increment which it would receive in a unit of time, if its change should be uniform throughout the interval. Thus, when a variable at a given instant is said to change at the rate of 2 inches per second, the meaning is, that an increment of 2 inches would be added in one sec- ond, if the change should continue uniform for one second. 2. If a circular plate of metal is expanded by heat, how rapidly does the area increase, when the radius is 2 inches long and increases at the rate of .01 inch per second ? Let X = radius, and y = area; then y = irx^, and dy = 2trxdx. When a; = 2 inches, and dx = .01 inch per second, dy = .04 tt square inches per second, which is the rate at which the area increases. 3. The common logarithm of 1174 is 3.069668. What is the loga- rithm of 1174.8, if the logarithm is assumed to change uniformly with the number ? ^ Let X — the number, and y = its logarithm ; then y = log x, and dy = ~ dx. Hence, the increment of the logarithm is — times as great as the increment of the number. Therefore dy = '^'^^^^^^ X .8 = .000295. . . . And log 1174.8 = 3.069668 + .000295 = 3.069963. Remark. It will be seen from the equation dy = — dx, that as the X number increases by equal constant increments, the logarithm will increase more and more slowly. So the assumption made in the last example is not strictly true, but for comparatively small changes in the O- CI /^= ^^-' ..4 r--' '-^-^ •^-r— ^— -^ . X ^ A , ' ■vC*j^ f cJT c-j A, m..—i 2Z~ a^ c / ' - f( DIFFERENTIALS. 39 number, the results are sufficiently accurate for practical applications. The use of the Tabular Differences in tables of logarithms is based on this assumption. 4. In the parabola y^ — 12 a?, find the point at which the ordinate and abscissa are increasing equally. Ans. The point (3.6). ■ 5. At what part of the quadrant does the arc increase twice as rapidly as its sine ? Ans. At 60°. 6. The logarithmic sine of 30° 5' is 9.700062. What is the logar rithmic sine of 30° 6' ? Ans. 9.700280. 7. A boy is running on a horizontal plane directly towards the foot of a tower at the rate of 5 miles per hour. At what rate is he approaching the top when he is 60 feet from the base, the' tower being 80 feet high ? Ans. 3 miles per hour. 8. A vessel is sailing northwest at the rate of 10 miles per hour. At what rate is she making north latitude ? Ans. 1.^1 -f- miles per hour. /t ^^ ^N ^ /'■-i ^"^ - r I ,^ ^ . / 'I -' "^ - y J < CHAPTER V. INTEGRATION. Art. 29. Definition. Integration is the operation of finding the function from which a given differential has been obtained. The result of the integration is called the integral of the differential. The symbol which indicates the operation of integration is I . Since differentiation and integration are inverse operations, the symbols d and j , as signs of operations, neutralize each other.* The process of integration is of a tentative nature, depending on a previous knowledge of differentiation; just as division in arithme- tic is a tentative process depending on a previous knowledge of multiplication. For example, d {x'^) = 4 y^dx ; therefore ( 4a^da; = cc^ /< Art. 30. Fundamental Formulas. The fundamental formulas for integration are obtained directly from the formulas for differentiation. A function is the integral of a differential, if the function when differentiated produces the differ- ential. All integrations must ultimately be performed by the formulas of this article. When a differential is to be integrated, if it is not apparent on inspection what function when differentiated produces it, * The symbol / is derived from the initial of the word "summation." Leibnitz introduced the letter S to denote the operation, and this gradually became elon- gated into the symbol J". 40 « VII. « VIII. a. INTEGRATION. 41 the differential must be transformed into some equivalent expression of known form, whose integral is given by one of the fundamental formulas. 1. C(du H- dv — dw) =:u + v — w'j from I. 2. ladv = av; " II. 3. I uav'''hlv = av"; 4. C^ = a\ogv; J V 5. r aMogadv = a"; ** IX. a. 6. |Vdv = e% « IX. 6. 7. I cosvd?; = sinv; « ^ XI. 8. i — sinvdv = cosv] " XII. 9. j sec-vdv = tanv; " XIII. 10. I — cosec^vdv = cotv; " XIV. 11. j sec V tan v dv = sec iJ ; " XV. 12. j — cosec'ycotvdv = cosecu; " XVI. 13. j sin ?; d?; = vers V ; ^ " XVII. 14. I ~ cos V dv = coyevsv'j " XVIII. 15. f ^^ = arc sin ^; " XIX. 16. f ^^ = arccos^; « XX. 42 17 DIFFERENTIAL AND INTEGRAL CALCULUS. from XXI. XXII. arc tan v; rs. I —^ = arc cot V : J 1 -\-ir Y dv ^ vVv^ — 1 r dv •^ v\^v^ - f 19 20 = arc sec v ; arccosecv: V2v ■/■ 22. I - dv arc vers 'y; = arc covers v ; XXIII. XXIV. XXV. XXVI. ^2v-v^ Art. 31. Elementary Rules of Integration. The first four rules of integration will be demonstrated in fulL (1) By I., d(u-\-v — w) = da + dv — dw ; hence I d(u-\-v — w) = I (du + dv — dtv), u + v — w=: I (du + dv — dw). u-{-v — w= I du-\- I dv— i dw, j (du + iv — dw) = I du-\- I dv— I dw. or But therefore Hence, the integral of the algebraic sum of any number of differen- tials is equal to the algebraic sum of their integrals. (2) BylL, d(av) = adv'j hence j d(av) = j adVf or av= j adv. But av = ajdv therefore j adv = aj dv. INTEGRATION. 43 Hence, the integral of the product of a constant and a differential is equal to the product of the constant and the integral of the differential. . (3) By VIL, dav" = nav^'-Hv. Then 1 dav" = 1 nav'^'^dv ; refore 1 nav''-Hv = av'^. Hence, when a function consists of three factors, — viz. a constant factor, a variable factor with any constant exponent except —1, and a differential factor which is the differential of the variable without its exponent, — its integral is the product of the constant factor, by the variable factor with its exponent increased by 1, divided by the new exponent. (4) ByVIII.a, da\ogv = a—' Then - Cda\ogv=C—', therefore | = a log v. Hence, the integral of a fraction whose numerator is the product of a constant by the differential of the denominator, is equal to the product of the constant by the Naperian logarithm of the denominator. Art. 32. Coj^stant of Integration. By III., it is seen that the differential of a constant is zero ; hence, constant terms disappear in differentiation. Therefore, in returning from the differential to the integral, some constant must be added, which is called the constant of integration. The value of this arbitrary constant is determined in each case after integration by the data of the given problem, as will be shown hereafter. So, for the present, the undetermined constant will be omitted, but its addition after each inte- gration will always be understood. Frequently, when a differential is integrated by different methods, the results may not appear to agree, 44 DIFFERENTIAL AND INTEGRAL CALCULUS. but on inspection it will always be found that the integrals differ only by some constant. ■♦ PROBLEMS. •* Formulas 1-3. 1. I aMx. ■/' U»^>A By Formula 3, making v = x, and n = 4, '4 <^ I' ^''^' CaMx = \C4.axHx = ^^' 2. Ch{Q ax^ + 8 bx^)i (2 aa; + 4 bx^) clx. diQax'-^S bx^) = (12 ax + 24 bx')dx', hence, if (2 ax -f Aba^) dx be multiplied by 6, it will be the differential of (6 aar^ 4- 8 5a;^)^ without the parenthesis exponent. After dividing the constant factor by 6 to preserve the same value, the integration may be effected by Formula 3, in which /v = 6 ao? -\- 8 b^x?. Therefore Cb (6 ax" + 8 by?f (2 ax + 4 bx") dx = f- (() ar", + 8 bx^) ' (12 ax + 24 6a^ dx ^(eax'-^Sbx^y^ = - -g = ^ (6 ax^ +86x^)1 3 3. f ^^^ = Ci(a' + x'')-hxdx=(a'-^a^i 4. C—^x~^dx. Ans. i^x'K 5. j (^ax^ — ^bx^)dx. Ans. ax^ — bx^. Jdx —^' Ans. 2-Vx. %. a.)^ «*-*! *-* <^ ~^*** ^1* ^^.~-**^aJ^^ fL^ X o^"*^ **-* .6;**-* 9xx = ***** i 1 0*-t-.0«* A^>^ «:^ **- -— »* <> *'**^' Tl^..-* /TT <»i2-»«a. «- «- ' "^ («— -<^ X -*- /. ^-;x^ .x'>'''o^''--'''^ ^"-C X li'^^ t.- <>• •t ( .. . ^ -*.*^ W^ MU.. .,^«u.^ .^ ^ ^^ ^^^^ lu^ ^.^^ ^ "ii^ <>l^*.'«A _^^ t.a.» ; INTEGRATION. 45 10. C{Zax' + Uy?f^{2ax-\-^h7p)dx, Ans. -^ (3 aa^ + 4 6a^)i 11 C ^^^ Ans _2aV36^T4^^ 13. Formulas 4-6 r &Mx 6 + 2a;3 .. 14 . fJ^. Ans. log (a; -a). */ a? — a ft-i 15. r^l^. ^ns. — log(a+6a^). ^^ r\xdx ^ Ans.lo^ix' + l)^. Jx^ + i * 17. r(loga;)3^. Ans. i(loga;)\ 18. ri^^. J:„«. log (3 a;* + 7) a. 19. fi^. - ^«.s. »-| + |-log(x + l). 20. /6a-d. = ^Ja-. loga.2cte = JJ-, by FormulaS.^^^^^ ^ ^ Se'^da;. Ans. 3e*. "^ «2. Cbe'^dx. Ans. -e*". J a 46 DIFFERENTIAL AND INTEGRAL CALCULUS. 23. jSa^'xloga'dx. Ans. fa< "24. fa'^dx. Ans. __^^ J 1 4- Inc l + loga Formulas 7-14. 25. 1 cos mxdx. /cos mxdx = — I cos mx-dmx = — sin ma;, by Formula 7. mj m ' '' 26 . fsin^ (2 a;) cos (2 a;) (^a;. fsin^ (2 a;) cos (2 a;) dx = ifsin^ (2a;) cos {2x) 2 da; = I- fsin^ (2 a;) d sin (2 a;) = | sin* (2 a;). 27. rsec2(a;3>)a^(^^ ^^^ J tan a;'. 28. j 5 sec (3 a;) tan (3 a;) (Za;. ^ns. | sec (3 a;). „o Tsin (3 a;) da; . , ^ 2^- I 0,0 X • ^^«- i sec 3a;. J cos2(3a;) ^ 30. 1 e'=°**sina;c?a;. ^ns. — e<=«''*. J(l + cos a;) da; a -i r , - -^ - — ' 7-^ — • Ans. loff ra; + sina;l. a; + sina; ^ ■■ I tan a; da;. Ans. log sec a;. 32. I tan a; da;. 33. j sin B sec'^ 6 dO. Ans. sec 0. 34. I cot a; da;. Ans. log sin a;. n^ C (l^ C ^^ r^sec^(^ a;) da; , . , 35. I-, — = 1^-7-^^ r-= I ^ . ^V — = logtanJa;. */ sin x J Z sm \ x cos \x J tan \x 36. f-^ = r f ^ = log tanf? + i»\ K. ^'>' J 6- a <^K t, K /^ ^C.^. ^^K S - - ^C-'^V .^ ---* f^ 3»7 fihr ii.V. * X. **«&!^ f^ Jb.il-,vx, > *jr. 37 INTEGRATION. 47 . I— Ans. log tan a;. J sm a; cos a; 38. I -7-T — ^— :; — Ans. tana; — cota;. J SI] 39. ■ ^^ '* J V^ 6V Formulas 14-22. • 6a; = - arc sm — a / da; ^ r g _ 1 r a ^ 1 In order to integrate the preceding differential by Formula 15, it must be transformed into an equivalent differential having unity for the first term under the radical sign, and having for its numerator the differential of the square root of the second term under the radical sign. 40. f ^^ . r dx _ r a _l r ^ ^"^ ^dx 1 hx = - arc sec — , by Formula 19. a a — Ans. .- arc vers — • 42. C_^l^_. Ans. arc sin (a;^ 43. r ^^ Ans. arc cos (2V^). *^ ^x — ^y? 44. r^^^. Ans. ^arctan(ar^. 45. J 8a; "^dx _ ^ ^^^ 4 V6 arc vers (6 a;*). ^2a;7_6a;* 48 DIFFERENTIAL AND INTEGRAL CALCULUS. 2dx 46. ( — -• Ans. arc cot -• J 4 + aH 2 ^'^- I r; — TT o' -4*^''^- arctan(a; — 1). J 2 — 2x-{- yr ■ ^ dx . 1 ex 48. I :z=z==z' Ans. -!^arcsec Vc-x2 _ ^2^2 ah ah r dx_ ^ Va^x — Aa I (^^ .1 2h^x 49. I • ' Ans. -arc covers — — • "6V h 2 V3* 50. I = I ; — - = — =arc tan (x + I) Art. 33. Integration of Trigonometric Differentials. Trigonometric differentials, to which the previous formulas cannot be made applicable by algebraic reductions, may often be brought to known forms by trigonometric reductions. Every function may be differentiated by a general method, but there is no general method of integration. Thus, while every function may be differentiated, but a limited number of differentials can be integrated. In attempting to integrate any given differential, the object is always to transform it to a fundamental form whose integral is known. Hence, the processes of the Integral Calculus are transformations to effect reductions to fundamental formulas. In order to become pro- ficient in these operations, it is necessary to have much practice in the solution of problems. PROBLEMS. 1. I eos'xdx. lcos^xdx= I (^ -^ ^cos2x)dx = ^x-\- \sm2x. 2. j sin^a;(?a;. Ans. Jic — Jsin2a;. 3 . I tan'^ X dx. itsai^xdx= j (sec^a; — l)tana;da; = ^tan'^a; — logsecic. INTEGRATION. 49 4. itsin^xdx. Ans. tana; — aj. 5 . I tan* X dx. Ans. ^ tan^ x — tan a; + a;. 6 . I tan^ x dx. Ans. \ tan* x — ^ tan^ x + log sec x. 7. I sin^xdx. isiu^xdx= I (l — cos^xysmxdx= j (— 1 + 2cos^a; — cos'*x)dcosa; , 9 o cos'' a; = — cos a; + I cos^ x 5 ' x dx. Ans. sin x — ^ sin^ x. 10 . I COS^; . j cos^ X dx. Ans. sin x — sin^ « + f sin^ ^ — t sin' x. . j cot* a; da?. Ans. — Jcot^a;4- cota; + a;. 11. rcos*a;da;= ("(^ + icos2a;)2(^a; = Ja; + isin2»4- i fcos^ (2 a;)c? (2a;) = J a; 4- ^ sin 2 a; -f- 1^ [a; + ^ sin 4 a;] = I a; + ^ sin 2 a; + -5^2 sill 4 a;. Art. 34. Definite Integrals. It was shown in Art. 32, that an arbitrary constant must be added after each integration. Before the value of this constant is determined, the integral is said to be indefinite. If, from the data of the given problem, the value of the integral is known for some particular value of the variable, the constant can be determined by substituting this value of the variable in the indefinite integral. For example, let ds — = gt -\-v', in which g and v' are constants. 50 DIFFERENTIAL AND INTEGRAL CALCULUS. By integrating and adding the constant C, *' Now if S = S' when t = 0, C will be equal to /S', and* S = igt^-{-v't-{-S'. If, in any indefinite integral, two different values of the variable be substituted for the variable, and the result given by the second substi- tution be subtracted from the first result, the constant of integration is eliminated, and the integral is said to be taken between limits. The definite integral of f'(x)dx between the limits a and b is indi- cated thus : '^y(x)dx, (1) JT in which a is the superior limit and b the inferior limit of integration. In (1), /'(ic) dx is first to be integrated, then a and b are to be succes- sively substituted for x, and the second result is to be subtracted from the first. It is assumed that the integral is continuous between the limits a and b. A function is said to be continuous between two values of the variable when it has a single finite value for every value of the varia- ble between the given values, and changes gradually as the variable passes from the first value to the second. Evidently, the value of the integral up to the superior limit includes the value of the integral at the inferior limit. Hence, the difference between the values of the integral at two limits will be the value of the integral between those limits. Assuming that the inferior limit is equal to b, and writing the inte- gral in the two different ways ; ff'{x)dx=f(x) + C, (2) and jy'(x)dx =f(x) - [/(x)], (3) If these two forms are taken to represent the same quantity, f(x) + C = f(x) - [/(a;)],; whence C= [/(x)],. (4) INTEGRATION. 51 Thus it will be seen that the upper limit is any final value of the increasing variable x, and that the lower limit may be assigned without defining the upper limit. Equation (4) shows that the constant C de- pends on the lower limit. Therefore, the integral in equation (2) Is indefinite because a free choice is left with regard to the selection of both limits. The part f(x) depends on the value of x selected for the superior limit, and the part C depends on the value taken for the infe- rior limit. Art. 35» Geometric Illustration of Definite Integration. The problem of finding the areas of plane curves was one of those that gave rise to the Integral Calculus, and this problem furnishes an illustration of the preceding article. In Fig. 7, let MN represent any plane curve ; it is required to find the area included between the curve, the X-axis and two ordinates. Let (aj, y) be the coordinates of the point F. If BS = Aa be added to Xf SQ = y + Ay. QC and FD are drawn parallel to the X-axis. If A represents the area of the curve between two ordinates and the X-axis, AA = area BFQS. Y1 C_0_- R S W Fig. 7. Then BCQS ^CR^ y-hAy ^. Ay BFDS FR y y' Now, as Ax approaches zero, Ay also approaches zero j and limit ^^^ = 1. FFDS 62 DIFFERENTIAL AND INTEGRAL CALCULUS. But the area RPQS is intermediate between area MCQS and area RPDS', hence RPDS ' or' limit -A4_ = i y ' ^x dA . or - —r = ^'^ yax therefore dA = ydx, and A==Jydx-ha (1) If the area between the ordinates NW and ML is required, the superior limit will be the abscissa W, and the inferior limit will be the abscissa OL. If these limits are respectively a and 6, the area will be denoted by * ,. A=JJydx. (2y Let the particular curve whose area is required be the common parabola, then y =: ^2px. Substituting this value of y in (1), gives A= CV2pxdx+0 = V2p Cx^dx-{- (7 = |V2pa;t + 0. If the area is estimated from the origin, when a; = 0, A = 0; hence, by the first method of Art. 34, (7=0, and, therefore, A = | V2 j9a;^. If the area is required between two ordinates whose abscissas are a and 6, by the second method of Art. 34, j^^/2^dx = [I V2^a;f]' = |V2^ [a^ - 6^]. Art. 36. Change of Limits. Let Jf\x)dx=f{x); then JJr(x)dx=f(a)--f(b) ;.:*.;.' ■ ■- ■ [/(&)-/(«)]■:;: '.:^ .-: =^-£f'(x)dx. - . . ^. . INTEGRATION. 53 Hence, the limits of integration may be interchanged by changing the sign of the integral. It may also be readily shown that j' f (X) dx =£ r ix) dx + jT' /' {X) dX. If a new variable be substituted for the old variable in integration between limits, corresponding changes must be made in the limits of integration. For example, I x'^dx is required. Suppose x = z^] then when x = 4, 2; = ± 2, and when x = 1, z = ± 1, Therefore f x'^dx = 2 f^ z^^'+^dz n + l PROBLEMS. 1. Find the particular integral of dy=(x^—b^x)dx, ii y=0 when x=2, Ans. y = t-^^2b'-4, 4 2 2. Find the particular integral of du = (1 -|- 1 ax) ^dx, if ?^ = when a;= 0. I ». jr(.--,...[=p-g;=l'. 2 r''l2 -4:e)de = 0.64. 8. f^^^M = V2 - 1. i 4 . r6T^dx = 3S. 9. p:^da; = 4 J2 Jo Va; */o a^ + a^ 2 a 'Jo ^^2 _ ^a, — \rT, f J2 l+ar" 2 Jo " 54 DIFFERENTIAL AND INTEGRAL CALCULUS. — . assume z—-- Ans. f . 13.^ In I sinxcos^ajdic, assume siiia; = 2!. Ans. \. 14. In C ^^^ . assume 2/ = l-ar'. ^ns. ± 1. 15. Find the area of the curve y — Q^-\-^Xy between the abscissas x=^^ and a; = 0. -4ns. 180. Note. Applications of the Integral Calculus in rectifying curves, determining areas, volumes, centre of mass, and moment of inertia, will be found in Chapters XIX., XX. and XXI. CHAPTER VI. SUCCESSIVE DIFFERENTIATION AND INTEGRATION. Art. 37. Successive Derivatives. As the derivative of a function is, in general, a new function of the independent variable, it can be differentiated. The derivative of the first derivative is called the second derivative. Likewise, when the second derivative is a function of the independent variable, it may also be differentiated, giving the third derivative; ana so on. For example, if y = ax*; •V. dx dx \dxj ^rd^fdf dx\_dx\clx^ = 24 ax, etc. The symbols for the successive derivatives are usually abbreviated as follows : d^fdy\^^ dx\dxj doc^^ dx\_dx\dxjj dx\d7?J da^ d /d"-VA ^ d"y dx\dx''-'^) dx"" The successive derivatives are often called successive differential coefficients. As the first derivative is often denoted by/'(ic), the suc- cessive derivatives are often denoted by/"(a;),/'"(ic), etc. 55 56 DIFFERENTIAL AND INTEGRAL CALCULUS. If differentials are employed, successive differentials will follow instead of successive derivatives. The differential obtained immedi- ately from the given function is the first differential ; the differential of the first differential is the second differential ; and so on. If the function be represented by ?/? the successive differentials will be denoted by dy, d-y, d^y, etc. In successive differentiation it is customary to make the assumption that the differential of the independent variable is constant ; i.e. the independent variable increases by equal increments, and hence is called an equicrescent variable. The independent variable will always be understood to be equicrescent unless the contrary is explicitly stated. , . Art. 38. Successive Integration. Two, three or n integrations must be performed in order to obtain the original function from which a second, third, or, in general, an nth derivative was derived. For example, if ^ = 24 ax. (1) Integrating (1), ^ = 12a3^: " _ (2) Integrating (2), ^ = 4aa^. .... . , .,, . - (3) Integrating (3), y — ax^. (4) But an arbitrary constant should be added after each integration, as a constant term may have disappeared at each differentiation. Then, in general, let — ^=/(a;); and denote the successive inte- grals of the function by /i(a;), /2(a?), /sCa;), etc., and the constants of integration by Cj, C2, C3, etc. «^-° S=-^(^)^ ■ then g^=/,(a;)+Q, SUCCESSIVE DIFFERENTIATION AND INTEGRATION. 57 d--'y_ dx--' --Ai^) + C,x + 02, d--''y_ dx--^ =fz{^) + C,'^+C^+C,, and finally, Q(f 1-1 .r"-2 I.2.3.. .(n-1) ' ^^1.2 3...(n- -2) PROBLEMS. 1. y = aaf. 2. 2/ = = tan X. ^^nax^' dx 1. > dx = sec^a;; %=<^- -1) aa;"-2; dx^ = 2 sec^ X • tana;; + C7„. ^^ = n(n - l)(n - 2) aaf»-« ; ^3,, • •• ••• ,_. .»=" ^ = k.a. ^ = 8tana;sec2a;(3sec2a;-l). da;" ■- dx^ ^ 3. y = ax^ + ha?. ' " — " . ' 4. ?/ = log (x + 1). • 5. 2/ = a*. ; - . : 6. ?/ = 6a;* — 4iB* — 6aj*. 8. ?/ = a;^ log (aj^. 9. 2/ = tan^a; + 81ogcosa;4-3a^. :0. d'y _ dx* :-6(a;4-l)-^ d-y^ dx- = a%togay. d'y _ dx'' = 144. ^y_ 24a;(l-ar^ d^ (1+aO* d^y da^ _48^ a; d'y dx' = 6 tan* a;. 68 DIFFERENTIAL AND INTEGRAL CALCULUS. 10. y = xe". i!y=:(x-\-n) e\ 11. ?/ = e*"°''«cos(a;siiia). ^ = €== '=<^'"» cos (a; sin a + a) ; , dx ^ = e '"« « cos (a; sin a + na). 13. ^ = 2x-8. 2/ = logx+C,^' + C2X + C3. 14. ^=cosa;. 2/ = cosx + Oi- + C,^ + C.x + 04. fl_^L_^Ur = 0. 16. ^ = .6366 a, when 2/ = a sin a;. 17. Find value of t from a^i^ = & (c - a;). de ^ ^ Integrating once gives 'dt^- a^=6(2ca;-ar^; hence ^^ = V? ^"^ Therefore t = -v/^arc vers -• 6 c -4' 18. In the harmonic curve whose equation is ^ = rj sin mZ + ^s cos m?, find —-; ri, rg and m being constants. ,„, ^^ Ans, ^ = -m%. ^ ^^f' APPLICATIONS IN MECHANICS. 69 APPLICATIONS IN MECHANICS. Art. 39. Velocity and Acceleration of Motion. The mean velocity of a moving body for a certain period, is equal to the distance passed over expressed in some unit of length, divided by the length of the period expressed in some unit of time. The velocity is uniform if equal distances are traversed in equal times; and the velocity is variable if unequal distances are traversed in equal times. Let s = distance, v = velocity, and t = time. And let As denote the increment of distance passed over by the body in the increment of time A^, while the velocity has increased to v -h Av. The distance actually passed over, if the velocity is variable, lies between the distances it would have passed over if its velocities at the beginning and end of the period had been uniform j hence vAt If (2) be differentiated, supposing y to vary and x to remain con- stant, the derivative is written |^ = ?>. (4) dy These derivatives are called partial derivatives. According to the differential notation, equations (3) and (4) may be transformed into -^dx = a dx, and -^dy -= bdy, ax dy and these expressions are called partial differentials. 64 l' ^ 2 ' FUNCTIONS OF TWO OR MORE VARIABLES. 65 Therefore; a partial differential of a function of several variables is a differential obtained on the hypothesis that only one of the variables changes. A total differential of a function of several variables is a differential obtained on the hypothesis that all of the variables change. To distinguish between the partial differentials of a function, the following notation is adopted : — Ax and — Ay will represent partial dz dz ^^ ^y increments, and ^-dx and —dy partial differentials of z, with respect dx dy to x and y, respectively. DxZ and [ —] have been used to represent total derivatives of z with respect- to X. The general equation of a surface as given in Analytical Geometry of three dimensions is in which x and y are independent variables. If the surface be cut by a plane parallel to the XZ plane, the equa- tion of the curve of intersection will contain the variables x and z only,- and the slope of the curve will be expressed by — dx Likewise, the equation of a section of the surface parallel to the YZ plane will contain the variables y and z only, and its slope will be — dy Art. 43. Total Differential of a Function of Two OR More Independent Variables. Let z=f(x,y). Let X and y be given successive increments Ax and Ay, and repre- sent the corresponding total increment of the function by Az, Let z'=f(x-^AXjy)', then ^ Ax =f(x + Ax, y) -fix, y), (1) Az' ^ Ay =f{x -{-Ax,y-\- Ay) -f{x + Ax, y), (2) Ay H ** 66 DIFFERENTIAL AND INTEGRxiL CALCULUS. and A2! =f{x + Ax, y -\- Ay) - f(x, y). (3) Adding (1) and (2), and placing the first member of "the resultant equation equal to the first member of (3), gives Ax Ay ^ ^ ^ Now, if Ax and Ay approach zero, limit Az' — limit Az, therefore, dz = —-dx-\- ^-cly. ox dy Hence, the total differential of a function of two variables is equal to the sum of its partial differentials. Similarly, the total differential of a function of any number of independent variables may be found to be equal to the sum of its partial differentials. PROBLEMS. 1. zz^aary^. ^dx=^3axFy^dx: —dy = 2ax^ydy. dx dy therefore dz = S aa^y^dx + 2 ax^y dy ; 2. z — T?. dz = yx^-^dx-\-af\ogxdy. 3. ^ = arctan?. dz = cos (xy) [^ydx-\- x dy"]. 5. 2=3 2/"^*. dz =:y'''"' logy cos xdx-\-^^^^dy. if 6. M = x". du= x^'~^(yzdx 4- zx logxdy -}- xylogxdz). Art. 44. Total Derivative of u with respect to x when ^ =/(a?, y, 2), y = (x), and z = 0i(aj)." By Art. 43, du = ^dx + ^-^dy-\-^dz', dx dy dz thpfp dM^du.dudy^.du dz^ « >. dx dx dy dx dz dx FUNCTIONS OF TWO OR MORE VARIABLES. 67 Cor. 1. If u =f(x, 2/), and y = (x), therefore • ^^Su^dudy_ ax ox ay ax ^ ^ Cor. 2. If u=f{y, z), y = (x), and 2 = <^(a;), dy dz .. therefore du^dud^_^eu^_ , dx dy dx dz dx - Cor. 3. If u =/(2/), and y= (x), du = — dy: dy ^' therefore du^dudy^ dx dy dx (4) In the proposition, u is directly a function of x and also indirectly a function of x through y and z. In Cor. 1, u is directly a function of x and indirectly a function of X through y. In Cor. 2, u is indirectly a function of x through y and z. In Cor. 3, u is indirectly a function of x through y. PROBLEMS. 1. u = e'"(y — z), y = a sina;, and z = cos x, du _du du dp du dz dx dx dy dx dz dx du ax/ ^ du _- du „ dy dz . dx dx 68 DIFFERENTIAL AND INTEGRAL CALCULUS. . therefore — = ae"' (y — z) -\- ae"^ cos x -\- e""" sin x dx ^^ = e''^{a^ sinx — acosa; + acosa; + sina:) = e"'' (a^ + 1) sin x. 2. w = arctan(a;i/), and y = e* du ^ e' (1 -\- x) ^ 3. u = yz, y = e% and z=x'^-^x^-\-12^-2^x+14.. ^ = eV. da; 4. w = log (r^ - 2/2)^ and 2/ = rsin^. ^ = -2tan^. dv 5. w = — )f-——l 2/ = a sin a;, and 2f = cos a;. — = e** sin a;. a^ H- 1 da; Art. 45. Successive Partial Derivatives of Two or More Variables. If u =f(Xy y\ then — and — are, in general, functions of both x ox oy and ?/, and may be differentiated with respect to either independent variable, giving second partial derivatives. The partial derivative of — with respect to x is — (—]z=—. dx dx\dxj dse^ The partial derivative of — with respect to w is — ( — ) = — • dy dy\dyj 5/ The partial derivative of — with respect to ?/ is — f — ] = — —. dx 6y\dxJ dydx ■ The partial derivative of — with respect to x is — ( —] = — —- ay dx\dyj dxdy Likewise, ^ is a third partial derivative, obtained by three suc- cessive differentiations ; first, with respect to x regarding y as constant, and then twice with respect to y regarding x as constant. dy\dxdyj dydxdy dx\dxdy^J dx^dy^^ and similarly with all other partial derivatives. IMPLICIT FUNCTIONS. 69 Art. 46. If u=f(x,y). to prove that ^ ^ = ^ ^ » On the supposition that x alone changes in/(a;, y), ^u ^ f(x + Ax, y) -f{x, y) A.^ Ao; Now, supposing y alone to change in (1), _A_ /Aw\ ^ /(a;4-Aa;, y-}-Ay)-f(x, y-{-Ay)-f{x-\-^y)-{-f(x, y) Ay\^xj Ai/ • Aa; On the supposition that y alone changes in /(ic, y)j ^u.^ fjx, y + Ay) -/(a;, y) Ay Ay Now, supposing x alone to change in (3), A Mm\ /(x+Aa;, y4-Ay)-/(a;4-Aa;, y)-f(x, y4-Ay)4-/(a;, y) . /^x ^x\AyJ AX' Ay ^ ^ (1) (2) (3) Equating (2) and (4), ^^(£)=A(^;). ' Hence at the limits, — f — )= — f — V dy\dxj dx\dyj In the same manner it may be proved that 5®M B^u d^u dx^ dy dy dx^ dx dy dx This principle may be extended to any number of differentiations, and to functions of three or more variables. Art. 47. Implicit Functions. When in f{x, y) = 0, y can be expressed as an explicit function of a;, the derivatives may be found by the methods already given. In tWs article a useful formula is established for obtaining the first derivative of an implicit function. Let i^=/(a.', y) = 0. (1) Then by Art. 44, Cor. 1, dx dx dy dx 70 DIFFERENTIAL AND INTEGRAL CALCULUS. But u = 0, and therefore its total derivative equals ; hence !^ + |^^ = 0. (3) ax ay ax ^ solving (3) for |, gives | = -| (4) For example take a^ + 2 2/a; + r^ = 0. Then u = a?-{-2yx + r^. p = 2x + 2y, p = 2x. ax ay Therefore by (4), ^ = - ^^±^ = - ^±^. ^ ^ ^' dx 2x X However, when an implicit relation between x and y is given from which y cannot readily be expressed as an explicit function of x, it is not necessary to resort to the method just given. But /(a?, y) = may be immediately differentiated with respect to x, treating t/ as a function of X, giving what is called' the first derived equation, from which —- dx can be obtained as a function of x and y. For instance, given : 07^ — 3 axy -|- 3/^ = 0. The first derived equation will be 3a^-3a2/-3aic^4-32/'^ = 0. (1) dx dx ^Solving (1) for ^, ^^ ^-ay ^ ^ ^ dx dx ax - f PROBLEMS. / , V .p a^u 5V 1. « = oos(x + 2,); ■ venfy— = ^-^. o — ^!_ihJ^. 'f ^^^ — ^^^ 2/^ — a^ ' dxdy dy dx 8. w = arc tan ( ?^ ) ; verify —^ = — -^. \xj dy^dx dxdy^ /J V I IMPLICIT FUNCTIONS. 71 5. w = sin (aa;" + ft?/") ; verify dx^ By dz dx^ dy^ dif dx^ 6. f-2m^y + af-a = 0. ^ = ??^^^, and g = ^M^. ax y — mx dxr {y — mxy 3 2/ + a; = 0. dy ^ 1 dx 3 (1 - 2/2)' Art. 48. Integration of Functions of Two or More Variables. Since integration is the inverse of differentiation, a partial deriva- tive is integrated by reversing the process of differentiation. j . For example, the integral of ■— =f(x, y) is found by integrating ox^ twice with respect to x, regarding y as constant ; but as y is regarded as a constant in this integration, it must be noticed that the constant of integration is an arbitrary function of y. d^u ■ ■' ' ' Again, let it be required to integrate =/fe y)- dydx^ This may be expressed Evidently, in the second differentiation, — - was differentiated with dx reference to y regarding x as constant ; therefore |=//(x,.)d,. (2) In (2), ^l is evidently such a function that its derivative with respect to a; is I f{x, y) dy ; 72 DIFFERENTIAL AND INTEGRAL CALCULUS. therefore, ^ ~ I I / (^? 2/) ^2/ U^^> or u =jjf{x, y) dydx. In Art. 46, it was proved that the values of the partial derivatives are independent of the order in which the variables are supposed to change, hence the order of integration is also immaterial. 5^ Similarly, if . eye:ey-^^^'y^' then u "^ffffi^y y) dydxdy; and if Jaler^^^'^'^^' then " ^SSSS^^'^' ^' *)'*^'*2/* ff PROBLEMS. 1. Cfy^'^^^-- 2 ■W- •■ £££-' Art. 49. Integration of Total Differentials of the First Order. If tfc be a function of x and y, by Art. 43, du = ^^dx-^^dy. (1) dx dy . ^ And from Art. 46, # ^^^ V # (^/)' (2) Therefore, if a total differential of a function of x and y is given of the form du = Pdx-^Qdy, (3) then P=^, and Q=^, ax oy Hence from (2) ^=^' (4) dy dx y^' CHANGE OF THE INDEPENDENT VARIABLE. 73 which is the condition that must be satisfied to make (3) an exact differential. This condition is called Euler's Criterion of Integrability. When (4) is satisfied, (3) is an exact differential of a function of x and y. Then the function u is obtained by integrating either term of (3); thus i^=Jpdx +/(?/). (5) In (5), the integration is with respect to x^ hence the constant of integration is an arbitrary function of the variable which is treated as a constant, and /(y) must be determined so as to make — = Q. by For example, let du = 2 x^f^dx + 3 x^yHy. Here P=2xif, and Q^Sx^f. Hence, ^=6xy\ and ^^Gxf. oy ox ^ Therefore (4) is satisfied, and (5) gives u= i 2 xy^dx = x^y^ +f{y) = ^1^ + c. PROBLEMS. 1. du = ydx-\-xdy. w = icy + c. 2 . du — ^ a^y^dx + 3 x*y^dy. u = a; V + c. 3. du = j-{-(2y-^^dy. u = ^-^y'-\-c. Art. 50. Change of the Independent Variable. Hitherto, the derivatives — , — , etc., have been obtained on the dx dx- supposition that x was the independent variable and y the function, but it is sometimes advantageous to change the function into another one in which y is made the independent variable and x the function. And occasionally it is desirable to make a new variable, of which both x and y are functions, the independent variable. 74 DIFFERENTIAL AND INTEGRAL CALCULUS. (a) To express — in terms of — • dx dy If 2/ is a function of x, then x may be regarded as a function of y, and*y may be treated as the independent variable. Evidently Ay Ax ^ Ax Ay and as Ay approaches zero, Ax approaches zero, and at the limit, ^x — =1- dx dy dy_l_ • ' ' • therefore dx ~ dx (1) dy ■ " (6) To express — ^ in terms of — and — -, also to express — in ^ ' rl^ fill rl-ifi •"■ /7/m3 dx" . „ o dx d^x -I d^x terms of — , --, and — -• dy dy^ dy^ From (1), By Art. 44, Cor. 3, dy dy^ d^ d^^±fdy\ d3? dx \dxj d^_± ( 1 dot? ~~ dx I dx therefore d 1 d 1 dx dx dy . ~dy dx dy) d'y d dx'~'dy 1 'dx dy} d^ dy^ dy dx dx dx dPx dy^ fdxV dx fdxV (^) CHANGE OF THE INDEPENDENT VARIABLE. 75 Similarly, ^ = d dx ~ d-x 1 df /dx\^ d dy r d'x ' dy' fdxV \dyj_ dy dx V ixVd'x ^fdxVfd'xY lyj df [dyj W) dy m dx [dyjdf W) _i fdx' / (3) In equations (1), (2) and (3), the independent variable is changed from xtoy. (c) To express -^, — ^, etc., in terms of -^, J, etc., when x is dx dx^ dz dz^ sdme given function of 2;. . . , By Art. 44, Cor. 3, dy _dy dz^ dx dz dx^ therefore ^ = -- ("^^ = — (^ — • dx^ dx\dxj dz\dxjdx^ and ^^±f^dz_ dx^ dz\dx^Jdx In equations (4), (5) and (6), the independent variable is changed from a; to 2;. (4) (5) (6) PROBLEMS. 1. Change the independent variable from a; to ^ in dx^ \dxj dx Substituting the values of ^„ and ^ from (1) and (2), dar dx 76 DIFFERENTIAL AND INTEGRAL CALCULUS. X therefore df_ dy d'x + 1 + ir-i=o: dx\ dx dy fdx\^ dy = 0. d/ \dy^ 2. Change the independent variable from a: to 2 in when X = cos z. By (4), and ^ ^dx^ dx ' dy _dy dz ^ dx dz dx^ ^ = -sin2, hence ^ = — A-> dz dx sins dy_ 1_ d^ dx sin z dz By (5), d^y _ d^ fdy\ dz ^ dx^ dz [dxj dx ' Therefore dz\dxj dz\ sin zdz J _ cosg dy _ 1 d^y sin- z dz sin z dz^ d^y _ _ /cos z dy _ 1 d^y\ 1 dar^ vsin^zc?2; - sinz c?2^/sin« — _ /" CQSZ d^ _ 1 dy \sin^ z dz sin^ ^ dz^ ) (7) (8) Substituting the values of -^ and -4 from (7) and (8) m the given dx dar example, gives -(1 Hence \sin^ z dz sin^ z dz^J \ sin z dz) ^y dz^ ra. CHANGE OF THE INDEPENDENT VARIABLE. 77 3. Given y =/(«), and x = F(t), to express -^, and — ^ in terms of ^, ^, % and ^. 5 («r dt^' at' "~" df B,A.t.44,Co.3, 1 = 11 (1) Differentiating (1), and treating -^ as a function of t through a?, gives dx ^_d^do^ dy d^ df~ da^df dxdt^' d^^dy^^ d^y dx fe dy „ d*w d^ dic df df dt df dt ,ox ^^^^^ S? = d^ = dl (^) dt;' df - ' CHAPTER VIII. , / DEVELOPMENT OF FUNCTIONS. Art. 51. Definition. A Function is said to be developed when it is transformed into an equivalent series. By the Binomial Theorem, constant powers of a binomial can be developed into series. For example, Some fractional functions may be developed by actual division. For example, l-3aj The Calculus method of development is a general method, including the developments just given and many others as special cases. This is one of the most important applications of successive derivar tives. Art. 52. Maclaurin's Theorem. Maclaurin's Theorem is a theorem by which a function of a single variable may be developed into a series of terms arranged according to the ascending integral powers of that variable, with constant co- efficients. The function to be developed is Assume the development of the form y =f(x) = A-^Bx-{-Cx'-^Da^ + E^-*- (1) 78 DEVELOPMENT OF FUNCTIONS. 79 in which A, B, C, D, etc., are constants to be found by the me,thod of Undetermined Coefficients, Forming the successive derivatives of (1) : ^=B-\-2Cx+ SDa^+ 4jE;a^+... (2) dx ^ ^= 2C+2'3Dx-h 3.4^x-2+- (3) ux^ ^= 2'3D -i-2'S'4.Ex -{-"' (4) Since (1) and consequently (2), (3), etc., are assumed to be true for all values of x, they will be true when » = 0. Hence, making x = in each of these equations, and representing what i/ becomes on this hypothesis by (y) ; what -^ becomes by f-^] ; what J becomes by /^2 \ dx \dxj dx^ ' — ^ J ; and so on : there follows from (1), (2/) = A, or A = (y) ; Substituting these values of A, B, C, ••• in (1), gives ,=/(.) = (,) + (g.,.(3)|V(g)|+... (5) If the function and its successive derivatives are expressed by f{x), n^), r{x), f'%x\ etc., equation (5) may be written, 80 DIFFERENTIAL AND INTEGRAL CALCULUS. • • y =/(^) =/(0)+/'(0) I +/"(0) I +/"'(0)| + -, (6) ^vilich is the formula of Maclaurin's Theorem.* A If in the attempted development of a function by Maclaurin's j Theorem, the function or some one of its derivatives becomes infinite when x = 0, the function cannot be developed by Maclaurin's Theorem. j This is evident, because a finite function cannot be equal to a series / containing infinite terms. ^ PROBLEMS. 1. To develop 2/= (<^ + a;)". Here f{x) =(a + a;)"; hence /(O) =a". f{x) =w (a 4- «)"-'; " /'(O) =wa"-^ fix) = n{n - l)(a + xy-^-, « /"(O) = n{n - l)a^-\ f"'(x)=: n(n- l)(n - 2)(a + xy-' ; " /'"(0)= n(n - l)(7i - 2)a' n-3 Substituting in (6), Art. 52, y=(a + xy=a^-^na^-'x-^ ''^''-^) a--'x^ + ''(''-}^(''-^K »-^a^-^..., \2 [3 which is the same development as that given by the Binomial Theorem. 2. To develop 2/ = log (1 + a;). . Here, f(x) = log (1 + ») ; hence /(O) = 0. I ^ M" -^ ^"^ =rT^' m r(^)=-7T-^^T-.; " /"W = (l+o:) ■2.m (1 +35)^' f"(x)= \' ^ • ^ : " /"(0)= 1.2m. * This theorem is commonly known as Maclaurin's, having been first published by him in 1742 ; but as it had been given in 1717 by Stirling, it should more prop- erly bear the name of the latter. DEVELOPMENT OF FUNCTIONS. 81 Substituting in (6), Art. 52, gives y = \og(l + x)=m(x-'^ + ^-^ + ...\ and if the logarithm is in the Naperian System, log(l+.)=x-f + f-}+-.. . Thus the logarithmic series is found to be a special case under Maclaurin's Theorem. 3. To develop y = sin x. f(x) = sin a; ; hence /(O) = 0. f'(x) =cosa;; " /'(O) =1. f"(x)=-smx', " /"(0)=0. f"'(x) = -cosx', " /'"(0)=-~l. Therefore, 3, = sina; = 05-^ + ^-^ + .... i£ [^ LL In the successive derivatives of sin a;, the first four values are periodically- repeating ; i.e., the fifth derivative equals the first, the sixth equals the second, etc.; hence, in general, d'^Csm x) _ gjn f ^ + w - V Obtain by Maclaurin's Theorem the following developments : ^ ^.4 ^6 4. COSiC = l— ; . ^ \± ^ The general formula for the successive derivatives of cos x is d"(cosa;) cos ( x + w ^- J ■ By the aid of the last two developments, natural sines and cosines may be computed. For example, to find the sine of 45°. By Art. 22, the circular measure of 45° is -• Substituting this value of x in the series of Prob. 3, gives = .7071068. 82 DIFFERENTIAL AND INTEGRAL CALCULUS. 5. a- = l + loga^ + log2a^4-log«a^4--. 1 ^ [^ If a = e and a; = l in this series, the value of the Naperian base may pe computed. /y»3 /ytO /yj 6. arctana; = aj — — + — — 77 H . o 5 7 The labor in finding the successive derivatives may sometimes be lessened by expanding the first derivative by some one of the algebraic methods, as follows : * f(x) =tan~^a;. By substituting cc = 1 in the development of arc tan Xj 4 3^6 7^ By Trigonometry, arc tan 1 = arc tan ^ + arc tan ^. "-i=[i-l©'-KIJ-]-[i-i(IJ*KiJ--} Therefore ir = 3.1415924-. /v«2 /y»4 n^ 8. e"°* = l4-a;+ --^ ^-.... 2 2-4 3-5 9. e*seca;=l4-a; + a^4-^H • o 11. arcsm^ = ., + 273 + 2:^ + 2:4:^ + -. It was by the use of this series that Sir Isaac Newton computed the value of v. DEVELOPMENT OF FUNCTIONS. 83 12. Develop y = logic. f(x) = log X ; hence /(O) = — oo. f'(x) = -; hence /'(O) = oo. f"(x) = - ^ ; hence f"(0) = - oo. Substituting in Maclaurin^s Formula, gives, y = logx = — cc + QO^-QO — H . In this example, \ogx equals a series of terms involving oo, which makes the development indeterminate for all values of x. Hence this function cannot be developed by Maclaurin's Theorem. 13. Develop y = cot x. 14. If e be substituted for a in Ex. 5, Substituting a; V— 1 for a;, . e^^~i==l — — -f -— — H uV^/^a; — — + — — — 4----^ [2+^ [6+ +^ ^'^ |3+[5 [7+ J = COS a; + V— 1 sin a;, by Exs. 4 and 3. (2) Substituting — x V— 1 for x in (1), gives similarly g-^^-^ = cosa;— V— 1 sinar. (3) Combining (2) and (3), gives sin a; = ^ — , 2V-1 and cosa; = '— These values of the sine and cosine are called their exponential values. The real functions, ^"^ ~ ^ "^ and ?f_+_l^, are called respectively the hyperbolic sine and hyperbolic cosine of x and are written sinh x and coshic. 84 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 53. Taylor's Theorem. * Taylor's Theorem is a theorem for developing a function of the sum of two variables into a series of terms arranged according to the ascending powers of one of the variables, with coefficients that are functions of the other variable. Taylor's Theorem depends on the following principle which must first be established : The derivative of a function of the sum of x and y with reference to x regarding y as constant, is equal to the derivative of the function with reference to y regarding x as constant. Let w=/(a; + 2/). Substituting z — x-{-y, gives u =/(.). In the first case, du_df(z) dz dx dz dx =/'(.), since 1^=1. In the second case, du _ df(z) dz dy dz dy =/'(2), since ^ = 1. dy Therefore du du dx = dy' This principle may readily be shown to apply in any particular example ; for instance, let u = (x-\-yy, then dx dy ^ Art. 64. Demonstration of Taylor's Theorem. Let u=f{x-\-y). Assume the development to have the form u=f{x^y)=A + By-^Cf-\-Df-^'*', (1) DEVELOPMENT OF FUNCTIONS. 85 in which A, B, (7, • • • are independent of y, but are functions of x. It is now required to find values of A, B, C, ••• by the method of un- determined coefficients. Differentiating (1), first with reference to x, regarding y as constant, then with reference to y, regarding x as constant, dx dx clx^ dx dx * ^=B + 2Cy-{-3Df-^4:Ef + .^.. dy But by Art. 53, — = — ; therefore, dx dy ^+^.y+^^^ + ^/ = 5 + 2 0?/ + 32)2/2 + 4^/+.... (2) dx dx dx dx ^ v / Making y = in (1), gives A =f(x). Since (2) is true for every value of y; equating the coefl&cients of like powers of y in the two members by the principle of Undeter- mined Coefficients, ^ = B, henceB = ^ . =/'(.), f = 2C, hence C = l|(/(.) =l/"(x), f=3A hence Z> = ||(i/"(.)) = i/"'(.); etc Substituting these values of A, B, C, ••• in (1), gives u=f{x-{- y) =f(x) +r(x) y +f\x) ^' +/"'(x) t + ..., (3) which is Taylor's Theorem. If x= be substituted in (3), it reduces to m =/(0) +/'(0) y + /-"(O) t +/'"(0) 2^ + ..., If 12 86 DIFFERENTIAL AND INTEGRAL CALCULUS. which is Maclaurin's Theorem. So Maclaurin's Theorem may be con- sidered as being but a special case of the more general one, Taylor's Tneorem.* PROBLEMS. 1. To develop (x -j- ?/)". Substituting y = 0, and taking the successive derivatives, f(x) =x% f'(x) =na;"-i, f"(x) =n(n-l)af'-^, f"'(x) = n(7i- 1) {n - 2) a^-3. Substituting these values in (3), (x + yy = x« + nx^-'y + ^(^~^) x^'Y + ^(^-l)(^-2) ^n-SyS + ...^ which is the Binomial Formula. 2. To develop sin (a; -f- 2/). /(aj) == sin X, f'(x) = cos x, f"(x) = — sin X, /'"(aj) = — cos x. Therefore sin (a; + 2/) = sin aj/^l — -| + -|- j + cos xfy — ^ -f ^ j = sin X cos y + cos x sin y. Obtain the following developments by Taylor's Theorem : S. a=^+y = a'(l + loga • ?/ + log^ a^+ log^a^^ -.) If. l£ 4. Iog(a; + 2/) = logaj-f ^-^^-f--^-.... 5V -ry; 8 t ^ 2 ar^ 3 a:^ 5. (a; + y)^ = a;^ -f Ja;"^y - ^aj'V + A^"V • ♦Taylor's Theorem is named from its discoverer, Dr. Brook Taylor. It was first published in 1715, in a book by Dr. Taylor entitled Methodus Incrementorum Directa et Inversa. DEVELOPMENT OF FUNCTIONS. 87 6. log(l + sm*) = x-| + |-^+-. 7. log sec (x + y) = log sec x + tan x-y -{■ sec^ x • ^ 4-sec^a; • tana;* ^.... 3 Art. 55. Rigorous Proof of Taylor's Theorem. In the demonstrations of Taylor's and Maclaurin's Theorems, it was assumed that the development would take place in a proposed form, and an infinite series was used without ascertaining that it was con- vergent. On account of these, as well as other objections, the method used is not altogether satisfactory. But, on the other hand, a rigorous investigation is necessarily complex and indirect. The proof which follows is one of the least difficult ones. The following proposition must be first established : If (^ (a;) = 0, when x=a, and also when x = h, and if <\> {x) and <^'(x) are finite and continuous between these values; then \x) will vanish for some value of x between a and h. The limit of — ^ = -^, and hence -^ will have the same sign as -^ Ao; dx dx , ^x Ay when Aa; is taken small enough. If y increases as x increases, t- will be positive, and if y decreases as x increases, — will be negative. So, dy if "T- is always positive between the two given values of x, ^ {x) would be constantly increasing, and if -^ is always negative between the two values of x, (ji (x) would be constantly decreasing ; but neither suppo- sition can be true, as <^ (x) vanishes at the two given values for x. Therefore, '(x) must change its sign between the two values, but a variable can only change its sign by passing through zero or infinity, and 4>'(x) remains finite by hypothesis; hence, <^'(a;) must pass through the value zero. Let f{x) and its successive derivatives be finite and continuous between x = a, and x = a -\- h. 88 DIFFERENTIAL AND INTEGRAL CALCULUS. Assume * (X) =f{a + X) -/(a) - xf'io) - |/"(a) ... -~f{a) - r^S, (1) in -vj^hich ■rf(a + k)-f(a)-hf{a)-^f"{a)... -|/"(«)1- (2) o_l!L+l| In (2), it is to be observed that E is independent of x. In (1), it is evident that cf>{x) = when x = 0, and when x = h. Hence, <^'(^) iii^st be equal to zero for some small value of x between and h. Represent this value by x^. Taking the derivative of (1) with respect to x, .t,'{x)=r(a + x)-f'{a)-xf"(a)- p'"{a) ... - -^/-(a) -fs (3) Ir |yi — 1 [n = 0, when x = x^. But (3) also vanishes when a; = ; hence there is some value of x between and ajj, for which <^"(a;) = 0. Continuing this process to w + 1 differentiations, -+\x)=p^\a + x)-Ii, for some value of x between and h, ''+\x) = ; let this value of x be Oh, where <1, therefore r^\a + ek) = R. (4) Equating the values of E in (2) and (4), and solving for /(a + h), f(a-\-h) = f(a)+hf'(a) + ^f"(a)-. +,-/"(«) + ,-^/"^^(a + ^/O- (5) Now, since the only restriction imposed on a was that it must be finite, a may have any value ; hence x may be substituted for a in (5), which gives f(x + h)=f(x)+hfXx)+^^f"(x) ... -^ffn(x) + -^r-\x+eh). (6) From ^0), Taylor's Theorem follows whenever the function is such DEVELOPMENT OF FUNCTIONS. 89 that by sufficiently increasing n the last term can be made indefinitely small.* Art. 56. Remainder in Taylor's and Maclaurin's Theorems. The last term of (6), Art. 55, -^ — -f'-^x + Oh), is called the re- mainder after n + 1 terms. For example, \etf(x) = (1 + ic)™, then by (6), Art. 55, (1 + ic)™ = 1 -\- mx-\ ^— ^x^ H- • • • + ^"^' [m(m- l)...(m -n)(l -f- ^a^r"""^'!. |n + 1 ^ -' /pn + l In this development, [??i(m — 1) ••• (m — n)(l + 6'a;) '""""*] is the remainder. ' If X is less than 1, the last term can be made indefinitely small by sufficiently increasing m. Hence, when x/ + k) from (2) in (1), gives fix -{'h,y-f- k) = f(x, y) + h ~f{x, y) + k -/{x, y) ox oy Similarly, f(x -\-h,y + k,z + l) = f(x, y, z) -f h -^/(x, y,z)-\-k |- /(aj, y,z)+l |/fe y, ^) 4- ir^'^/C^'^ 2/, 2) + 3 « ^/(a^, 2/, ^) + ..• 1+ And in like manner a function of any number of independent vari- ables may be expanded. CHAPTER IX. EVALUATION OF INDETERMINATE FORMS. Art. 58. Indeterminate Forms. A function of x is indeterminate when the substitution of a par- ticular value for x gives rise to one of the following expressions : ^, ^,00-00, 0°, 00°, 1^*. CO The true value of a function which becomes indeterminate is the value which the function approaches as its limit, as the independent variable approaches the particular value which makes the function indeterminate. For example, to find the true value of — ^^ when x = a. X — a When £c = a, this fraction assumes the form -• If a + h is substituted for x, the fraction becomes (ci-^hr a -{- h — a = 2a-^h. Now if h approaches zero, the independent variable approaches the particular value a, and the function evidently approaches 2 a as its true vahie. Again, if both numerator and denominator of the fraction ^ ~^ X — a are divided by x — a, the quotient is x -{- a, and now when x = a, the true value is found as before to be 2 a. As another example, ar 91 92 DIFFERENTIAL AND INTEGRAL CALCULUS. By rationalizing the numerator, a — Va^ — ^ _ X XT a- - (g^ - x") ^ r 1 n ^ \_ By algebraic and trigonometric transformations the true values of some indeterminate forms can be readily found, but the Differential Calculus furnishes a method of very general application. Art. 59. Functions that take the Form -• Let /(a?) and <^{x) be two functions, such that f(po)= and (a) Let X take an increment h ; then by Taylor's Theorem, .. „ f(^)+f'i^)j^+f"(^)^+f"'i^)^+- f(x + h) ^ [2 \3 (1) Substituting a for x, making f(a) = and <^ (a) = 0, and dividing both terms of the fraction by h, f{a + K) ^ l£ i£ m. Hence, as ^ approaches zero, by Art. 5, (a) '{a)' ■ ^' fix) which is the true value of Vr^ when x — a. (x) If /'(a)=0, and <^'(a)=0, then '-^^ = ^, and the result is still indeterminate. In this case, dropping the first term of the numerator EVALUATION OF INDETERMINATE FORMS. 93 and also of the denominator of (2), dividing both terms of the fraction by — , as h approaches zero, \d) be not 0, the true value is 0. If /"(a) be not and "(a) = 0, the true value is x. PROBLEMS. ar* — 1 1. Find the true value of -, when x = l. x — 1 f{x)=a^-l, {x)=x-l', hence, f(x)=ba^j and '{x) — 1 ; therefore 4^ = -Q^ = ^ = ^y when a; = 1. <^{x) '(x) 1 2. Find the true value of , when x = 0. or 4,(x) ^ 0' ' fM =lzi^^ = whence = 0; £M=?iM =2, whenc. = 0; 4,"{x) 6 a! 0' ' rM^^J^ =1, wlien=. = 0; (x) J_ 0' EVALUATION OF INDETERMINATE FORMS. 95 Therefore, by Art. 59, 1 d f'l \ 4,'(x) f{x) _i>{x)^ dxKj. (x)J [ ix)J ^'{x) lf(x)J fix) dx{j\x)J lf(x)J and when « = a, /W^^MIZW; (1) hence tMl^fM. (2) 4,(0) <#.'(«) ' Therefore, the true value of the indeterminate form — can be found by the same method as that of the form -. However, in dividing (1) by :il— i, it vyas assumed that Zi^ is not equal to 0(a) 0(a) or 00 . But (2) gives the true value in these cases also, as may be shown as follows : ^ . Suppose the true value of ~Ar to be 0, and let ;i be a finite quantity ; then 0(a) (PCa) {a) ^ f'(a) + h{a) V(a) ' therefore /(a)^/(a). .. 0(«) 0'(«) Similarly, if the true value of Z^^ = qo v^hen jc = a, then JV^ = 0, and the 0Ca) f(a) same demonstration applies. ^ -^ ^ ^ Art. 61. junctions that take the Forms x oo and oo — x. Let f(x) X (x) = X (a)=^ = l', therefore, the true value may be found as in Art. 59. 96 DIFFERENTIAL AND INTEGRAL CALCULUS. Again, let f{x) — (t>(x) = cc — co, when x = a. The expression in this case can be transformed into a fraction, * which will assume either the form - or — , and the true value is found *^ as before. For example, to find the value of sec X — tan x, when cc = -• 2 sec X — tan x = = -, when a; = -• cos X 2 Therefore Ml ^[^.^^1 _o. Art. 62. Functions that take the Forms 0®, oo^, and 1±*. Let f(x) and <^ (x) be two functions of x, which, when x = a, take such values that [/(»)]*'' is one of the assumed forms. Let y = lf(xW^', then log 2/ = <^ (x) logf{x). (1) 1st. When/(aj) = oo or 0, and (x) = 0, (1) becomes (x)\ogf{x) = 0{±y,). 2d. When /(a?) = 1, and <^(x) = ± oo , (1) becomes c}>(x)logf(x) = (±oo)xO. Therefore, the true values of the logarithms of all the functions which take the forms, 0*^, oo°, and 1-°^, may be obtained as in Art. 61. For example, to find the value of af when x = 0. Let y =zxi'] then log 2/ = ic log a; = (— 00 ), when a; = 0. Hence, log^/ = ^^ = --, when a; = 0. dx^ ^ therefore log af = 0, when a; = ; hence af = 1, when a? = 0. C-L EVALUATION OF INDETERMINATE FORMS. 97 PROBLEMS. Find the true values of the following functions : Ans. 0. Ans. 00. Ans. 0. Ans, 1. Ans. 0. Ans. a. Ans. — 1. 1 ^• Ans. 1. Ans. J. Ans. ^. Ans. 1. Ans. 1. u4ns. e. Ans. e*. ^ns. 1. 1. loga;^ when a; = 00 . 2. tanic 3x' when x = ^'n: 3. 1 — \ogx when a; = 0. e' . 4. log tan 2 X log tan X when a; = 0. 5. .T" log Xy when a; = 0. 6. 2'sin|, when a; = 00. 7. sec a; ic since — - , when a; = ^. 2 8. 2 1 ar^-1 x-1' when a; = 1. 9. X 1 when a; = 1. log aj log X 10. cosec^ ^ ~ ;;2» when a; = 0. 11. 1 12. 2 1 when a; = 0. sin^ X 1 — cos X /IN tan X when X = 0. 13. a;"", when x = 0. 14. («"+l)", when a; = 00. 15. (-?• when a; = 00. 16. sin a;**", when a; = ^. (01- DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 63. Compound Indeterminate Forms. ♦ When a given function can be resolved into factors, one or more of which become indeterminate for a particular value of x, the true value may'be obtained by getting the true value of each factor separately. When the true value of any indeterminate form is found, that of any constant power of it can be determined. PROBLEMS. This may be put in the form — • —j J. ~\~ Qu J. — Xi in which the second factor only is indeterminate. {x) 1-x^' then -^-M = - f'\ = ^af-'= ^ when a; = 1. 'x — bx^-^ b b Therefore | '^ — ^^ | = — • 2. (fllzilt^E!^, whenc. = 0. (e* - 1) tan^ x ^ / tan x V {f - 1) a^ \ X J X P_^l=l, andr^l::il]=l; L ^ Jo L ^ Jo therefore ^^ — ~ ; an a; _ ^^ \f\^Q^ a; = 1. 3. log(l + x + a^ + loga-^+'^^ ^hen r. = 0. sec X — cos X 4. (ar^ — a^ sin 1^ , ' £^, when x — a. a^cosg^ 5. a;"* (sin a?) ^'"'f J 7 ^/Y , when a; = J. \2 sin 2xj 2 Ans, 1. Ans. 1 a'' Ans. 2»»+3 4*f, CHAPTER X. MAXIMA AND MINIMA OF FUNCTIONS. Art. 64. Defixitioxs ax^d Geometric Illustration. A maximum value of a function is a certain value at which the function changes from an increasing to a decreasing function. Or, in other words, f{x) is a maximum for that value of x which makes f{x) greater than f{x + li) and fix — li) for very small values of li. A minimum value of a function is a certain value at which the function changes from a decreasing to an increasing function. Or, f{x) is a minimum for that value of x which makes f{x) less than f{x + h) and f(x — h) for very small values of h. In Fig. 8, let the curve AB be the locus of y=f(x). Then PN represents a maximum ordinate, and P'T a. minimum or- dinate. As X increases toward OiV, y approaches a maximum value, Fig. 8. PN, and the tangent to the curve makes an acute angle with the X-axis. At the point P, the tangent line is parallel to the X-axis. Immediately after passing P, the tangent makes an obtuse angle with the X-axis. But by Art. 27, the slope of the tangent line is equal 99 K 100 DIFFERENTIAL AND INTEGRAL CALCULUS. to f(x) ; hence f\x) is positive before, and negative after, a maximum ordinate. Likewise it may be shown that f'(x) is negative before, and positive after, a minimum ordinate. Thus, f\x) = 0, at both maxi- mum and minimum ordinates. Therefore, a condition for both maxima and minima is — = 0. dx Art. 65. Method of Determining Maxima and Minima. For a maximum value of the function, f'{x) = and f{x) changes sign from + to — when x passes through a value corresponding to a maximum value of the function. For a minimum value of the func- tion, f\x) = and f'{x) changes sign from — to + when x passes through a value corresponding to a minimum value of the function. Hence, the roots of the equation f\x) = are first obtained. If a is a root of this equation, a value slightly less, and then- one slightly greater than a, are substituted for x in f\x). Let li repre- sent a very small quantity. If /'(a — h) is +, and f(a + ^) is — , then /(a) is a maximum. ' If f\a — /i) is — , and /'(a + 7i) is -}-, then /(a) is a minimum. For example, let y = h-\-{x — a)*. ay *-^ ^ ,V.—7A» "' Then -^ = 4 (aj — a)^ = : heno « ic = a. "^ dx Substituting a — h for ic, gives dx Substituting, a. + h for x, gives dx Here, f\x) changes sign from — to -h at x = a\ hence, a is the value of X which gives a minimum function. Therefore, y = b, sl mini- mum. By reference to Fig. 8, it will be seen that P'^'W is a minimum ordinate, and the tangent to the curve at this point is perpendicular to MAXIMA AND MINIMA OF FUNCTIONS. 101 the X-axis. In this case f'{x) changes sign by passing through oo . Any variable can change its sign only by passing through or oo , but it does not necessarily follow that there is a change of sign whenever f'(x)=0, or f'(x)=cc . At point P^^, the tangent is parallel to the X-axis, hence f'(x) = ; but f'(x) is -f immediately before and after reaching this value. Therefore, the values of x which make /'(x) = do not always give maxima and minima, so they are simply called critical values, or values for which the function is to be examined. It is evident also that a function may have several maxima and minima, and a minimum value may be greater than a maximum value of the same function. Art. 66. Conditions for Maxima and Minima by Taylor's Theorem. Let /(a;) have a maximum or minimum value when 0?= a. ^ Then if h be a very small increment of x, by Art. 64, f(a) >f(a -h h), and f(a)>f(a — h), for a maximum, also f(a) 0. However, if /"(a) ^ 0, then the sign of the second members of (1) and (2) will depend on the terms containing /'"(a), and since the terms containing /'"(a) have opposite signs, there can be neither a maximum nor a minimum unless /"'(a) also vanishes; and if /'"(a) = 0, then /(a) is a maximum when /^'^(a) is negative, and a minimum when /^^(a) is positive ; and so on. Rule: Find f'{x), and solve the equation, f'(x) = 0. Substitute the roots of this equation for x inf"(x). Each value of x ivhich makes f"(x) negative will make f(x) a maximum; and each value which makes f"(x) positive will make f(x) a minimum. However, if any value of x also makes f"(x)= 0, substitute this value in the successive derivatives until one does not reduce to 0. If this be of" an odd order, the value ofx will give neither a maximum nor a minimum; but if it be of an even order and negative, f(x) will be a maximum, if of an ■eveji order and positive, f(x) will be a minimum. The solution of problems in maxima and minima is often simplified .by the aid of the following principles : ^ 1. If any value of x makes af(x) a maximum or minimum, a being A positive constant, that value will make/(ic) a maximum or minimum. Hence, a constant factor may be omitted. J 2. If any value of x makes [/(a?)]" a maximum or minimum, w being a positive constant, that value will make/(.T) a maximum or minimum. MAXIMA AND MINIMA OF FUNCTIONS. 103 Hence, any constant exponent of the function may be omitted; or if the function is a radical, the radical sign may be omitted. 3. If any value of x makes log /(a;) a maximum or minimum, that value will make f{x) a maximum or minimum. Hence, to find a maxi- mum or minimum value of the logarithm of a function, the function only need be taken. 4. If any value of x makes f{x) a maximum or minimum, that value will make — - — a minimum or maximum. Hence, when a function is a maximum or a minimum, its reciprocal is a minimum or a maximum. 5. If any value of x makes a +f{x) a maximum or minimum, that value will make f{x) a maximum or minimum. Hence, a constant term may be omitted. • Each of the preceding propositions may be readily proved. For example, in (1), the first derivative of af{x) when placed equal to zero, will give an equation whose roots are the same as the roots of the equation formed by placing the first derivative of f{x) equal to zero ; hence, the critical values will be the same in both cases. PROBLEMS. 1. Find the maximum and Bainimum values of 2 ana mmin ar»-3a^-9a; + 5. Let 2/ = a^-3a^-9a; + 5; then ^ = 3a^_6a;-9. dx Placing the first derivative equal to zero, and finding the roots, 3a;2_g^_9^Q. therefore a; = 3 or — 1. The second derivative is — ^ = 6 a? — 6. 104 DIFFERENTIAL AND INTEGRAL CALCULUS. When x = 3, —^ = 12, and as this value of x makes the second de- rivative positive, it corresponds to a minimum value of the function. When x=-l, ^=-12, and this result being negative indicates a maximum. Substituting these values of X in the function, gives, when a; = 3, y =— 22, a minimum, and when x = —l, y = 10, a maximum. These results may be illus- trated graphically by construct- ing the locus of the equation. In Fig. 9 it will be seen that there is a maximum ordinate corresponding to the abscissa Fig. 9. — 1? and a minimum ordinate corresponding to the abscissa 3. Remark. It will be very instructive to construct the loci of the equations in the first few examples. Examine the following functions for Maxima and Minima : 2. 2/ = ar^ — Sic^-fSic' + l. Ans. x=l, gives a Maximum, 2 ; x = 3, gives a Minimum, — 26, 3. 2/ = 2 ar' - 21 a;2 + 36 » - 20. Ans. x = 1, gives Max., - 3; x = 6, gives Min., — 128. 4. ?/ = 3 ar^ - 9 a.-^ — 27 a; -f- 30. Ans. x = -l, gives Max., 45 ; X = 3, gives Min., — 51. 5. y = ^^ ~^)^ Ans. x=\a, gives Min., JJ a\ (X — Z X MAXIMA AND MINIMA OF FUNCTIONS. 105 Ans. This function has no real Max. or Min. 7- y 1 -\- X tan X 8. y = xi. 9. y sma; 1 + tan X 10. y = sinx (1 + cos x). 11. y::=(x-iy(x-^2)\ Ans. x = cos X, gives a Max. A71S. X = e, gives Max. A71S. X = 45°, gives Max. A71S. X = -7 gives Max. Ans. ic = — 4, gives Max. ; x=l, gives Min. ; x = — 2, gives neither. GEOMETRIC PROBLEMS. 12. Determine the maximum rectangle inscribed in a given circle. Assume an inscribed rectangle as in Fig. 10. Let the diameter CB = cl, and the side CD = x ; then AC = -VilF^li^'. Denoting the area by A, then A/ ::^B A = x^d- — x^, which is to he a maximum. By Art. 66, 2, the function x\d' - af) may be used. Fio. 10. Put Then Now X^((P _ r^^^ y dy dx ^=2d'-12x'' = 2d', whena; = 0: = 2d'x-4.x'' = 0', liencea; = 0, ora; = dV}- = — 4 c?^, when x = c^VJ. Therefore, x = c? V^, which is the side of an inscribed square, will give the maximum rectangle. 106 DIFFERENTIAL AND INTEGRAL CALCULUS. 13. Find the greatest cylinder which can be inscribed in a given right cone with a circular base. In Fig. 11, let CH be a cylinder inscribed in the cone OAB. Given AM= a, and 0M= h. Let NC = X, and NM= y. Denoting the volume by FJ then F= ■7r:(?y. Fig. 11. From the similar triangles AOM and CON, h_h-y a X , hence a ,=«(.- ■y)- Therefore, V^= t-^(^ — 2/)V> which is found to be a maximum when y =z^h. Therefore, the altitude of the maximum inscribed cylinder is one-third of the altitude of the cone. 14. Find the maximum cone which can be inscribed in a sphere whose radius is r. In rig. 12, let ADB and CDB be the semicircle and triangle which gene- rate the sphere and inscribed cone by revolution about AB. Let CD = x, CB=^y, and F= the volume of the cone ; then V=l7roi^y. ar^=03x CA = y{2r-y), hence, V= ^7ry^(2r — y), which is the function whose maximum is required. Ans. The altitude of the Max. cone = |r. 15. Determine the right cylinder of greatest convex surface that can be inscribed in a given sphere. If r = the radius of the sphere, and x = the radius of the base of the MAXIMA AND MINIMA OF FUNCTIONS. * 107 cylinder, then the convex surface of the cylinder is 4 ttx Vr* — a?. This will be a maximum when the radius of the base is V2 16. From a given surface S, a cylindrical vessel with circular base and open top is to be made, so as to contain the greatest amount. To find its dimensions. Let a;= radius of base, y= altitude, and V= volume of a cylinder. Then 4 F^rfy, (1) and S = -rrx' + 2 7rxy. (2) Differentiating (1) and (2) with respect to x : From (1), dx dx hence dy^_2y^ dx X From (2), = 27ra; + 27ra:^4-2 dx hence dy _ x-{- y^ dx X Hence 2y x-\-y X X Therefore 2/ = a;, or the altitude = radius of base. In this example, (2) might have been solved for y and this value substituted in (1), and the solution would have been the usual one. But the given solution is in this and similar examples much shorter. 17. What is the length of the axis of the maximum parabola which can be cut from a given right circular cone, knowing that the area of a parabola is equal to two-thirds of the product of its base and altitude ? Given BC = a, and AB = h, in Fig. 13, Let CM=x, then BM = a — X, and RS = 2V(a — x)x. 108 DIFFERENTIAL AND INTEGRAL CALCULUS. By similar triangles, a:h::x'.MN\ .-. MN=-x. Fig. 13. Hence, the area of the parabola is A = --x \/{a — X) Xy which is a maximum when x = \a. 18. What is the altitude of the maximum rectangle which can be inscribed in a given segment of a parabola ? In Fig. 14, let -BOO be the parabolic segment and AO=h. Let OH=x, MAXIMA AND MINIMA OF FUNCTIONS. 109 then MH = ^2px. Therefore, area of MRSN = 2 ■V2px(h — x), which is a maximum when a; = — 3 19. What is the maximum cylinder that can be inscribed in an oblate spheroid whose semi-axes are a and b ? Ans. Radius of base = J a V6 ; altitude = -f 6 V3. 20. Find the maximum right cone that can be inscribed in a given right cone, the vertex of the required cone being at the centre of the base of the given cone. Ans. The ratio of the altitudes is J. 21. What is the maximum isosceles triangle which can be inscribed in a circle ? Ans. An equilateral triangle. 22. What is the altitude of the cone of maximum convex surface that can be inscribed in a sphere whose radius is 3 ? Ans. Altitude = 4. 23. When is the difference between the sine and the cosine of any angle a maximum ? Ans. When the angle = 135°. 24. If the strength of a beam with rectangular cross-section varies directly as the breadth, and as the square of the depth, what are the dimensions of the strongest beam that can be cut from a round log whose diameter is Z) ? Ans. Depth = D V|. 25. A rectangular box with a square base and open at the top, is to be constructed to contain 108 cubic inches. What must be its dimen- sions so as to contain the least material ? Ans. Altitude = 3 inches ; side of base = 6 inches. 26. What is the altitude of the minimum cone that may be circum- scribed about a sphere whose diameter is 10 ? Ans. Altitude = 20. 27. A person, being in a boat 3 miles from the nearest point of the beach, wishes to reach in the shortest time a place 5 miles from that point along the shore ; supposing he can walk 5 miles an hour, but can row only at the rate of 4 miles per hour, required the place he must land. Ans. One mile from the place to be reached. 110 DIFFERENTIAL AND INTEGRAL CALCULUS. 28. Find the minimum value of y when y — Arts. 2/ = 0.32218. .29. Determine the greatest rectangle which can be inscribed in a given triangle whose base = 2 & and altitude = a. Ans. A = \ah, 30. A Norman Avindow 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. Ans. Radius of semicircle = height of rectangle. 31. A privateer must pass between two lights A and B on opposite headlands, the distance between which is c. The intensity of light A at a unit's distance is a, and the intensity of light 5 at a unit's dis- tance is h. At what point must a privateer pass the line joining the lights so as to be as little in the light as possible, assuming the princi- ple of optics, that the intensity of a light at any point is equal to its intensity at a unit's distance divided by the square of the distance of the point from the light ? a _ ca^ a^ + h^ 32. The flame of a candle is directly over the centre of a circle whose radius is 5 ; what ought to be its height above the plane of the circle so as to illuminate the circumference as much as possible, sup- posing the intensity of the light to vary directly as the sine of the angle under which it strikes the illuminated surface, and inversely as the square of its distance from the same surface ? Ans. Height above circle = 5 VJ. 33. If the total waste per mile in an electric conductor is C^r-\-- r (r ohms resistance per mile), due to heat, interest, and depreciation, what is the relation between C, r and t when the waste is a minimum ? Ans. Cr = t. 34. In the formula, A^B = p'(^ + ^^^^^^^ it is required to find the se^ 4- X* ' value of X that makes the variable factor — — — - a minimum. (a; - ly Ans. a; = 2.2. MAXIMA AND MINIMA OF FUNCTIONS. Ill 35. From the differential equation, 20,000,000 ^ = -100a;, find clx- the equation of the curve and the maximum ordinate; the first con- stant of integration being found by making ^ = when x equals I, dx and the second constant of integration being found by making y equal to zero when x equals zero. Ans. y —■ ( Y and Max. ordinate = ^ 20,000,000^ 2 6 / 600,000 36. A statue whose height is 10 feet stands on a pedestal 8 feet in height, which rests on a level surface. At what point on the horizontal plane through the base of the pedestal does the statue subtend the greatest angle ? Ans. 12 feet from centre of base. 37. If -u denotes the velocity of a current, and x the velocity of a steamer through the water against the current, what will be the speed most economical in fuel if the quantity of fuel burnt is proportional to a;^ ? Ans. x = ^v. Art. 67. Maxima and Minima of Functions of Two Indepen- dent Variables. Let f(x, y) represent any function of two independent variables. When fix, y) >f(x + h, y + k), for all very small values of h and k, positive or negative, the function has a maximum value. When f{x, y) , it may be proved that the sign oi f{x-\-h, y •\-k) —f{x, y) will depend on h ;r-+^"^i ^"^^ 112 DIFFEREXTIAL AXD INTEGRAL CALCULUS. therefore will change sign with li and A:; hence, a maximum or mini- mum value is impossible unless dx dy And since h and k are independent, |i* = 0, and 1^ = 0. (2) ax dy ^ ^ Substituting A = -—-, B = —^-, and 0= — -, in equation (1), srives dx^ dxdy dy^ \ y^ ^ fix J^h,y + k) -fix, 2/) = 1 {Ah' 4- 2 Bhk + Ck^) + ... In (3), the sign of fix + h, y + k) —f(x, y) will depend on ^| + i?Y+(^0-J5^1, and in order that it may retain the same sign for all very small values of h and A:, it is necessary that AC — B- should be positive; for if AC — B^ be negative, it will be possible, by ascribing some suitable value to - to make the whole expression change its sign. Hence as a condition for a maximum or minimum, B^ 0, and C > 0. -^ Therefore, the conditions established are : For either a maximum or minimum, £^ = 0, ^ = 0, and (^^\^^?1. dx 'By ' \dydxj ex's/ MAXIMA AND MINIMA OF FUNCTIONS. 113 Also, for a maximum, — - < and — - < 0, and for a minimum, t"^ > ^ ^^^ -r-.>^- Art. 68. Conditions for Maxima and Minima of Functions of Three Independent Variables. By an investigation similar to that of Art. 67, the following condi- tions for a maximum or minimum value of u =f(x, y, z) are established : For either a maximum or minimum. dii dx ' dy ' dz ' \dx dyj dx^ dy^^ \dydzj dy^ dz^' \ d^u \^ d\ d\ dz dxj dz^ da? Also, for a maximum, — - < 0, — - < 0, and — - < 0, dxr 62/ dz^ ->0, _>0, and - and for a minimum, — - > 0, — - > 0, and -^^ >0. PROBLEMS. 1. Find a point so situated that the sum of the squares of its dis- tances from the three vertices of a given triangle shall be a minimum."* Let (Xi, 2/1), (^''2? 2/2) s-nd. (% 2/3) be the coordinates of the vertices, and (x, y) the given point. Then, [(x - x,y + (2/ - 2/i)T + [{X - x,y 4- (2/ - y,y2 + [(^ - ^^3)^ + (y- ys)"] is the function to be a minimum, which may be represented by u. ^ = 2(x-x,)+2(x-x,)+2(x-x,), g = 2(2/-2/0 4-2(^-2/0 +2(2/ -2/3), * See Byerly's Diff. Calc, p. 236. 114 DIFFERENTIAL AND INTEGRAL CALCULUS. dxdy Making -^ and -^ each equal to zero : dx oy 2{x-x^-\-2(x- X2) ■j-2(x-x^) = 0, therefore a; = ^i-±^?-±^. 3 2(2/-2/0+2(2/-2/2)+2(2/-2/3) = O, therefore v = ^i + ^2 + ^3. ^ 3 ^O-52 = 36-0>0, ^ == 6 > 0. Hence, w is a minimum when and the required point is the centre of gravity of the triangle. 2. Find the maximum value of afy^ (6 — x — y). Ans. Max. when x = 3, y = 2. 3. Find the maximum value of 3 axy — a^ — y^. Ans. Max. when x== a, y = a. 4. What is the triangle of maximum perimeter that may be in- scribed in a given circle ? ^ ^ ., , , , Ans. An equilateral triangle. 5. Find the values of a;, y and z, that make oi^-\-y^-\-z^-\-x — 2z — aw a minimum. Ans. » = - 2, 2/ = - i, 2 = 1. MAXIMA AND MINIMA OF FUNCTIONS. 115 6. What rectangular parallelopiped inscribed in a given sphere has the maximum volume ? Ans. A cube. 7. An open vessel is to be constructed in the form of a rectangular parallelopiped, cax3able of containing 108 cubic inches of water. What must be its dimensions to require the least material in construction ? Ans. Length and width, 6 inches ; height, 3 inches. 8. Prove that the point, the sum of the squares of whose distances from n given points situated in the same plane shall be a minimum, is the centre of mean position of the given points. CHAPTER XI. TANGENTS, NORMALS AND ASYMPTOTES TO ANY PLANE CURVE. Art. 69. Equations of the Tangent and Normal. Let y =f(x) be the equation of any plane curve AB, and (x', y') the coordinates of the point of tangency T, in Fig. 15. The equation of a Fig. 15. straight line through T is y —y' = a(x — ic'), in which a is the tangent of the angle which the line makes with the X-axis. dy^ dx' By Art. 27, tSLJUj/ Therefore y-y dy' dx ■Xx-x') • (1) is the equation of the tangent to the curve y = f(x) at the poinlf (x'y y^).- As the normal to the curve at the point T is perpendicular to the dx' tangent at that point, the slope of the normal is -, and hence the dy' equation of the normal is y-y' dx' . ^ -(x—x'). 116 (2) TANGENTS, NORMALS AND ASYMPTOTES. 117 Art. 70. Lengths of Tangent, Normal, Subtangent and Subnormal. In Fig. 15, let TM be the ordinate, TR the tangent and TN the normal, at the point of contact ; then MR is the subtangent and MN the subnormal. Dir TM ,dx' RM= = V — -: tan MRT ^ dy^' hence hence hence hence dx' subtangent = v' dy' MN=MTtSinMTN=y'^: dx' subnormal = y'—; ' dx' RT=^V(MRy-\-(MTy = -yJ(y'^'+y''; tangent = y =V'^ '2 _L.f yl (]?r\. dx'J ' TN=V{MTy + {MN) normal = 2/'.Jl+^ (3) (4) (5) (6) Art. 71. Tangent of the Angle between the Radius Vector AT Any Point of a Plane Curve and the Tangent to the Curve at that Point, in Polar Coordinates. Let be the pole, OX the initial line, and P any point of the curve AB, in Fig. 16. Let (r, 0) be the coordinates of P, and (r + Ar, B -\- A^) be the coordinates of another point R of the curve. If PS is perpendicular to OR, then PAS' = rsinA^, and SR=(r-\- Ar) — r cos Ad. rsin Ad Therefore \>?,TiSRP = r + Ar — r cos Ad 118 DIFFERENTIAL AND INTEGRAL CALCULUS. Now, if the point R approaches P, then the secant EP approaches the tangent PT', and the angle /iSi^P approaches the angle OPT, If the angle OPT is represented by , then Fig. 16. tan<^ = limit r sin A^ r + A?' — rcosA^ r sin A^ = limit AO 2_ Ar A^ A^ mw, limit^ = l; limit ^ = ^*; 2sin2 limit A^ limit A^ 2 Therefore tan <^ = r d6 dr (1) Art. 72. Derivative of an Arc. In Fig. 17, let P and P' be two points on the curve AB separated by a short distance As. The coordinates of P and P' are (x, y) and (x 4- AXj y + Ai/) respectively. TANGENTS, NORMALS AND ASYMPTOTES. In the right triangle PMP\ hence As Aa; A«-^i+/A?/^^ Aic Y / F / Ax Ay M — / A X y X Fia. 17. Now, when P' approaches P, limit ^ = limit Jl + W, Aa; ^ \Aa;y or ds dx =V'H2)' 119 Art. 73. Derivative of an Arc in Polar Coordinates. From Fig. 16, regarding the limiting triangle of PSR, limit sec PBS = limit ^ = limit — , BS Ar hence therefore sec<^ = ds dr (1) I = VI + tan^ <^ =V^+r^^|J, by Art. 71, (1) ; ds_dsdr dO~drde 120 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 74. Lengths of Tangent, Normal, Subtangent, Subnormal AND Perpendicular on Tangent, in Polar Coordinates. In Fig. 18, let PT be the tangent to the curve at the point P, and thrdugh P the normal PN is drawn. NT is drawn through the pole 0, perpendicular to OP, PT is called the 2?o/ar ton^eni; PiV, the poZar iiormal; OT, the polar subtangent; ON, the polar subnoi'mal; and OD, the perpendicular on tangent from the pole. Or= subtangent = OP tan OPT= Opfr—\ by Art. 71, dr 0N= subnormal = OP tan OPiY= OP cot OPT _ dr (1) (2) TANGENTS, NORMALS AND ASYMPTOTES. 121 PT = tangent = ^'OW+OT' = ry^l + '^(j^' (3) PN = normal = VOW+OW' = ^r^ + f—Y- (4) (5) 0Z> = perpendicular = OP sin OPD tan OP/; 7^ Vl + tan- OPD PROBLEMS. ^-KS)' 1. Find the equations of the tangent and normal to the circle, a^ + 2/2 = r2. By differentiation ^ = _?;... ^' = _ ^'. dx y clx' y' Substituting this derivative in Art. 69 (1), gives y-y' = - -,{^ - ^'), whence, xx' -f- yy' = r^, which .is the equation of the tangent. Substituting ~^| = _ ^| in Art. 69 (2), gives y - y'= ^^(x - x% cix y X which is the equation of the normal. 2. Find the equations of the tangent and normal to the ellipse aV + b-x' = a-b\ 3. Find the equations of the tangent and normal to the parabola y^ = 2 px. 4. Find the equations of the tangent and normal to y^ = 2a^ — a:^ at x = l. Ayis. Equations of tangent : y = ^x -{- ^, y z= — :^x — ^. Equations of normal : y = — 2x + S, y = 2x — 3. 5. What is the inclination of the tangent to the curve a^if = a\x + y), at the origin ? Ans. 135°. 6. What is the value of the subtangent to 2/ = a'' ? Ans. m. 122 DIFFERENTIAL AND INTEGRAL CALCULUS. 7. What is the value of the subnormal to i/" = a^'^a ? Ans. -^» nx 8. At what angle do the curves y = — — -— ^ and 0^ = 4: ay intersect ? Ans. 71° 33' 54". 9. Find the values of the normal and subnormal to the cycloid a; = 2 arc vers | — V4 y — y^^ at the point where y = l. Ans. Normal = 2 ; subnormal = V3. 10. Find the values of the tangent, normal, subtangent and sub- normal to the spiral of Archimedes, whose equation is r = ad. Ans, Subtangent = - , subnormal = a, a tangent = r\l + — , normal = Vr^ + a*. 11 . Find the values of the subtangent and subnormal to r'^ = a* cos 2 $. Ans. Subtangent = — a^ sin 2d' a subnormal = sin 2 d. r Art. 76. Rectilinear Asymptotes. An asymptote to a curve is the limiting position of the tangent when the point of contact moves to an infinite distance from the origin. Hence, any curve will have an asymptote when the point of con- tact of a tangent is infinitely removed from the origin, and when the tangent intersects either coordinate axis at a finite distance from the origin. From Art. 69 (1), Intercept on X=x'-y'—, (1) dy' dv' and intercept on F=y' — a;'-=^- (2) ttX TANGENTS, NORMALS AND ASYMPTOTES. 123 If in (1) and (2) the intercept on X or F is finite when a' = oo or y' = 00, then the tangent at {x', y') is an asymptote. For example, to examine the hyperbola aV — b^^ = — a^b'^ for asymptotes : Here ^'=^'. dx' a^y' Hence intercept on X= x' f-^ X = - = 0, when x' = oo. a:' Intercept on Y= y' -— = ^, = 0, when y' = oo. ay y' Hence, there is an asymptote passing through the origin. dy' b'x' b 1 b , zn = -T-t = ± , = ± -» when ic = oo. dx' cry a ^2 a X/1- x^ Therefore, there are two asymptotes whose slopes are ± -, and the equations of the asymptotes are y = ± - x. If, when a;' = CO in (1) and (2), the intercepts on both X and Yare infinite, the curve has no asymptote corresponding to ic' = 00. If when y' = cc in (1) and (2), the intercepts on X and Y are infinite, the curve has no asymptote corresponding to ?/' = 00. If both intercepts are zero, the asymptote passes through the origin, and its direction is found by evaluating -^ for aj = 00. Art. 76. Asymptotes Parallel to an Axis. When a; = 00 in the equation of a curve gives a finite value of y, then there is an asymptote parallel to the X-axis. For instance, if y= a when ic = 00 in the equation of the curve, then y = ais the equa- tion of an asymptote, because it is the equation of a straight line passing within a finite distance of the origin, and touching the curve at an infinite distance. Likewise, when ?/ = 00 gives a; = 6 in the equation of a curve, then a; = 6 is an asymptote parallel to the F-axis. 124 DIFFERENTIAL AND INTEGRAL CALCULUS. For example, taking the curve whose equation is ]f- = X — b Here, y = go when a? = 6 ; hence x — b = is the equation of an asymptote to the curve parallel to the y-axis. Art. 77. Asymptotes determined by Expansion. An asymptote may sometimes be determined by solving the equa- tion of the curve for y and expanding the second member into a series in descending powers of x. For example, to examine y"^ x — b Here ^s^x^/'l-^'j \ hence y= ±x (l — ] as X approaches oo, (1) approaches y=±{^+l} (2) Hence, as x increases, the curve (1) is continually approaching the straight line (2), and (1) and (2) become tangent when a; = (» ; there- fore, y = ±(a;-f--j are the equations of two asymptotes to the curve (1). Art. 78. Asymptotes in Polar Coordinates. If a polar curve has an asymptote, as the point of contact is at an infinite distance from the pole, and as the tangent line passes within a finite distance from the pole, the radius vector of the point of contact is parallel to the asymptote, and the subtangent is perpendicular to the asymptote and finite. [See Fig. 18, Art. 74.] Hence, for an asymptote, the polar subtangent r^— is finite for r = 00. Therefore, to examine a polar curve for an asymptote, a value of is found which makes r = x ; if the corresponding polar subtan- TANGENTS, NORMALS AND ASYMPTOTES. 125 gent is finite, there will be an asymptote, and if the subtangent is in- finite, there is no corresponding asymptote. For example, to examine the hyperbolic spiral rO = a for asymp- totes. If r = 00 in r = -, then ^ = 0. e f = -^, hence 7-f = -a. dr IT dr Therefore there is an asymptote parallel to the initial line which passes at a distance a from the pole. PROBLEMS. Examine the following curves for asymptotes : Arts. Asymptote, yz= — x, Ans. No asymptotes. 1. y^ =z a^ — Qc^. 2. The circle, ellipse, and parabola. 3. 2/^ = ax^ -f- a^. 4. The cissoid, y^ = 5. f = 2ax^-x\ .3 2r-x a" 6. 2/ = c+, .^2 (x — by 7. The lituus, rS^ = a. Ans. y = a;-f-. Ans. x = 2r. Ans. 7 = — a; -}- I a. Ans. y = c, and x = h. Ans. The initial line. 8. r cos ^ = a cos 2 6. Ans. There is an asymptote perpendicular to the initial line at a distance a to the left of the pole. CHAPTER XII. DIRECTION OF CURVATURE. POINTS OF INFLECTION. OF CURVATURE. CONTACT. RADIUS Art. 79. Direction of Curvature. A curve is concave towards the X-axis at any point, when in the immediate vicinity of that point it lies between the tangent and the X-axis. A curve is convex towards the X-axis when the tangent lies between the curve and the axis. Fig. 19. In Fig. 19, let the coordinates oi P he (x', y'). The curve being concave downward, the ordinates of the curve for the abscissas x' ± h, h being a very small quantity, must be less than the corresponding ordinates of the line tangent to the curve at P. Likewise, in Fig. 20, the curve being convex downward, the ordi- nates of the curve for the abscissas x' ± h must be greater than the corresponding ordinates of the tangent line at P. 126 DIRECTION OF CURVATURE. 127 In either case, let (x' + h, y") be the coordinates of P'. If y =f(x) is the equation of either curve, then y" =f(x' + h), and Fig. 20. The equation of the tangent at P is If the coordinates of Pj are (aj' + ^, 2/2)? Now 2/2=/(^')+/'(^')/^; hence 2/" - y^ =/"(^')i^ +/'" (aj')|? + or if ^ be taken sufficiently small, ^ A' ^ 2/"- 2/. =/"(»') I (1) (2) (3) (4) In (4), 2/" — 2/2 will have the same sign as /"(a;'); therefore, the curve is concave to the X-axis if /"(a?') is negative, and convex if /" (a;') is positive. If the curve is below the X-axis, ?/" and y^ are negative, and the 'Curve is convex towards the X-axis when — y" -{- y2 is negative, that is, 128 DIFFEREl^TIAL AND INTEGRAL CALCULUS. iff"{x') is negative, and tlie curve is concave towards the Jt-axis when /" (x') is positive. ,Art. 80. Direction of Curvature in Polar Coordinates. A curve referred to polar coordinates is concave to the pole at any point, when in the immediate vicinity of that point it lies between the tangent to the curve at that point and the pole. A curve is convex to the pole when the tangent lies between the curve and the pole. If p is the perpendicular distance from the pole to the tangent to the curve at a point whose coordinates are (r, ^), it is evident from Fig. 21, that when the curve is concave to the pole, p increases as r increases : hence, -^ is positive. dr Fig. 21. Fig. 22. Similarly, from Fig. 22, when the curve is convex to the pole, p decreases as r increases ; hence ^ is negative. dr If the equation of the curve is given in terms of r and By the equar tion may be transformed into an equation between r and p by aid of Art. 74 (5) ; then the curve is concave or convex, according as -^ is positive or negative. POINTS OF INFLECTION. 129 Art. 81. Points op InfMction. A point of inflection of a curve is a point where the curvature changes from concavity to convexity or the reverse. Hence, at a point of inflection the curve cuts the tangent. By Art. 79, when /" (x) < 0, the curve is concave to the X-axis, and convex when /"(a;)>0; therefore f"(x) changes sign, and hence f" (x) =0 or 00 at a point of inflection. For example, to examine y = — for points of inflection. d^ 6x-2a ^^ hence x = ^. dx" b' ' 3 If a;<|, theng<0; 3 dor and if x>-, then ^,>0. 3' dx" Hence, f"{x) changes sign at the point whose abscissa is % and therefore this will be a point of inflection. PROBLEMS. 1. Find the direction of curvature and point of inflection of y=a+{x — bY. Ans. There is a point of inflection at (b, a) ; on the left of this point the curve is concave, while on the right it is convex. 2. Examine y = x + 36x^ — 2 x^ — x^ for points of inflection. Arts. At x = 2, and x = — 3. 3. Find points of inflection and direction of curvature oi y = — X -\- xZ A71S. (— 6, — I), (0, 0), (6, I); convex on the left of the first point, concave between first and second points, convex between second and third, and concave on the right of third point. 4. Find the direction of curvature of the lituus r = — Ans. Concave towards the pole when r a^2. 130 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 82. Curvature. The total curvature of a curve between two points is the total change of direction in passing from one point to the other, and is measured by the angle formed by the tangents to the curve at the two points. The actual curvature of a curve at a given point is the rate of change of its direction relative to that of its length.* Fig. 23. In Fig. 23, let P be any point of the curve AB. The angle PTX= ip, or the angle which the tangent at P makes with the X-axis is the direction of the curve at P. Likewise the angle P'SX is the * Leibnitz defined the curvature of a curve at any point as the rate at which the curve is bending, or the rate at which the tangent is revolving per unit length of curve. RADIUS OF CURVATURE. 131 direction of the curve at P'. Then angle TCS = A»/^ is the difference of these inclinations, and if PP' = As, and the point P' approaches P, limit ^ = *^, As cZs which is an expression for the curvature. If the curvature is uniform ; that is, if AB is the arc of a circle whose radius is r, the angle TCS = angle PMP' at the centre subtended by the arc PP', ppi and angle TCS = arc ; r hence A\f/_ _1. As r clip 1 ds r therefore _r = _. (1) ds r Hence, the curvature of a circle is equal to the reciprocal of its radius. Art. 83. Radius of Curvature. The curvature of a circle varying inversely as its radius, and as any value at pleasure may be given to the radius, it follows that there is always a circle whose curvature is equal to the curvature of any curve at any point. The circle tangent to a curve at any point and having the same curvature as the curve at that point is called a circle of curvature ; its centre is the centre of curvature, and its radius is the radius of curvature at that point. Denoting the radius of curvature by p, by Art. 82 (1), P=-' (1) Now it is required to find p in terms of x and y. By Art. 27, tan.// = ^-S^; hence sec^ i^r?i// dx dx ' (^l d^ therefore ^^=_^ = ^ (2) » ^ secV n_(^^Y 132 DIFFERENTIAL AND INTEGRAL CALCULUS. Substituting in (1) the value of dif/ just found and ds = dxyjl + f^^] from Art. 72, \dx P = ['Ht)'j dip d^ daf (3) which is the required radius of curvature. If I represents the length of the curve, equation (3) may be reduced dl^ to the formula p = —. dx ' dy Art. 84. Radius of Curvature in Terms of Polar Coordinates. Formula (3), Art. 83, is first transformed, any quantity t being taken as the new independent variable. The values of -^ and —^ from (2) dx dor and (3) of Ex. 3, Art. 50, being substituted in (3) of Art. 83, putting t = 0, dx^ dy^ ^ d^ dx _ d^ dy d&' de^ dO^'dO de'dO P = dx^ d^ da^ df\l dO' dOy d^ d^y dx _ d^x dy d$^"dd~d^' dO (1) From the equations of transformation, x = r cos $, and y = r sin 0, by differentiation, dx • /I . A dr | = -.cos.-2sin.| + cos.g, g = -.sin. + 2cos.| + sin.g. CONTACT. 133 Substituting these values in (1), [^-S] (2) which is the required formula. Art. 85. Contact of Different Orders. Let y=f{x) and y = (x) be the equations of any two curves re- ferred to the same axes. Giving to ic a small increment h, and expanding, f(x+h) = f(x) +f{x)h+r{x)^+r\x)^ + ..^, (1) ^(xJrh) = <^{x) + '(x)h + i>"(^) I + ct>"'(^) I + -. (2) If the two curves have a common point whose abscissa is a, then f(a) = '(d), the curves have a com- mon tangent ; this is called contact of the first order. If, also, f'\x) — "(x)y the two curves have contact of the second order. In general, two curves will have contact of the nth. order at a; = a, when the following conditions are satisfied : f(a) = <^ (a), f'(a) = <^'(a), f"(a) = cf>"(a), ... f%a) = <^"(a). If the curves have a common point at a; = a, and if a be substituted for X in (1) and (2), and (2) be subtracted from (1), then /(a + h)- From (3), a = x dx 1^,'dy dy dx d^ da^ (6) Substituting values of (y — h) and (x — a) from (5) and (6) in (2), after reducing, .tM. dx- The values of a and h in (5) and (6) are the coordinates of the centre of the osculating circle, and the value of r in (7) is its radius. Hence, by comparison with Art. 83, it will be seen that the osculating circle is the circle of curvature and the radius of the osculating circle is the radius of curvature. Art. 87. The Osculating Circle has Contact of the Third Order where the Radius op Curvature is a Maximum or Minimum. If p is to be a maximum or minimum, by Art. %Q, ^ = 0. dx Differentiating (3) of Art. 83, e^p 2^ dx^J d x\dx^J doi^\ dx^J _^ . dx^ jm ' ^' \dx'J therefore d?y dce[^ ^2) dx^ 14.^ dx" 136 DIFFERENTIAL AND INTEGRAL CALCULUS. The same value of — ^ as found in (2) will be obtained by differen- t^ating (5) of Art. S6. Therefore, the given curve and the osculating circle have the same value for -^ at a point of maximum or minimum , dxr curvature ; hence the contact at such a point is of the third order. PROBLEMS. 1. Find the radius of curvature of the parabola y^ — 2px at any point, and the coordinates of the centre of curvature. dy__p d^y _ p^. dx y dx^ y^ hence p = ±-^ ^ - , - , r p' p'' ^ ~Sx+p. 1 + "' / _f p^ At the vertex, a; = and 2/ = ; therefore /o = p, a = p and & = 0. 2. Find the radius of curvature of the ellipse aV 4- ^^y? = a}h^. 3. Find the curvature of 2/ = a;* — 4 ar' — 18 a^ at the origin. Ans. p = ^V 4. Find the radius of curvature of the cycloid a? = r arc vers ^ r -V2?-2/-/. Ans. p = 2V2ry. PROBLEMS. 137 X X 5. Find the radius of curvature of the catenary y = -(e'*-{-e "). Arts, p = — . 6. Find the radius of curvature of the spiral of Archimedes, r = a9. 2or -{-'T 7. Find the radius of curvature of the cardiod r = a (1 — cos B). Ans. p = f V2 ar. 8. Find the radius of curvature of the ellipse whose axes are 8 and 4, at a; = 2, and the coordinates of the centre of curvature. Ans. p = 5.86 ; a = .38, and 6 = - 3.9. 9. Find the order of contact of x^ — 3x- = 9y — 27 and 3 x = 2S — 9 y. Ans. Second order. 10. What is the order of contact of the parabola Ay = a^ — 4: and the circle x^ -\-y^ — 2y = 3? Ans. Third order. 11. What is the radius of curvature of the curve 16 ?/^ = 4 a;* — a^, at the points (0, 0) and (2, 0) ? . Ans. p = 1, and p = 2. 12. What is the radius of curvature of the curve y = oi^-\-5oiy^-\-6x, at the origin ? Ans. p = 22.506. 13. Find the radius oF cur vaturF and the coordinates of the centre of curvature of the curve ?/ = e*, at a; = 0. Ans. p = 2V2, (a, b) = (- 2, 3). CHAPTER XIIL EVOLUTES AND INVOLUTES. ENVELOPES. Art. 88. Definition of Evolute and Involute. The evolute of any curve is the curve which is the locus of the centres of all the osculating circles of the given curve ; the given curve with respect to its evolute being called an involute. ^ .P^ A Fig. 24. In Fig. 24, let AB be the given curve, and the centres of curvature of P', P", P'", etc., be respectively Pj, Pj, Pg, etc.; then the curve MN, which is the locus of Pj, Pj, P3, etc., is the evolute of AB.. Art. 89. Equation of the Evolute. The equation of the evolute is the equation which expresses the relation between the coordinates of the centres of all the osculating 138 EVOLUTES AND INVOLUTES. 139 circles of the involute. The values of -^ and —. derived from the dx dar equation of the curve are substituted in equations (5) and (6) of Art. 86, giving two equations, which, together with the equation of the given curve, make three equations involving ic, y, a and b ; by combining these equations, eliminating x and y, a resulting equation will be obtained showing a relation between a and b, the coordinates of the evolute, which is the required equation. Fig. 25. For example, to find the equation of the evolute to the common parabola, y^ = 2px. Here ^^l^^^^l. dx y da^ Substituting in (5) and (6) of Art. S6: ' 6^3,-^. ^ = -i!!i hence 2/» = f,*6i. P y y p" 3 (2) (3) The values of y^ and x in (2) and (3) substituted in the equation of the parabola, give p^b^ = 2p ^ ~^ ; therefore ^'=i-/^-py (4) 140 DIFFERENTIAL AND INTEGRAL CALCULUS. Equation (4). is the equation of the evolute. This evolute is called the semi-cubical parabola. Constructing the evolute, its form and position is as shown in Fig. 25, where OA = p. Q • If the origin is transferred to A, the equation becomes b^ = a^ 27p or as a and b are the variable coordinates, the equation may be written, - ^ 27p The semi-cubical parabola is so called from the nature of its equation ; the equation being solved for y gives the function expressed in terms of the variable with an exponent of three halves. Art. 90. A Normal to Any Involute is Tangent to its Evolute. Let P', in Fig. 24, be any point of the involute, whose coordinates are x' and y', and let (a, b) be the coordinates of P^ the centre of curva- ture. Then the equation of the normal at P' by Art. 69 (2), is y-y' = -'^,(^-^')- (1) As (1) passes through (a, b), x'-a+^,(y'-b) = 0. (2) Now if P' moves along the curve, Pj moves along the evolute ; hence a, b and y' are functions of x'. Differentiating (2), dx' dx'' ^^ ^ ^ [dx'J dx' dx' ^ ^ But since (a, b) is on the evolute, by Art. 86 (5), 6 = 2/' + d'y' dx"' (.'-6)g+i+i:;=o. (4) EVOLUTES AND INVOLUTES. 141 Substituting (4) in (3), • _da _ dj/_ db_ _ a . _ ^ _ ^, dx' dx' dx' ' dy' da Hence, equation (2), which is the equation of the normal to the involute at (x', y'), may be written which is the equation of a tangent to the e volute at the point (a, b). Art. 91. The Difference between Any Two Radii op Curvature OF AN Involute. The equation of the circle of curvature at (x', y') is (x'-ay-{-(y'-by = p\ (1) Differentiating (1), y', a, b and p being functions of x', gives (.._.)_(.._.)g+(y_,)|;_(,._,)g=,g. (2) By Art. 90 (2), x' - a -\-^,iy' -b)=0; (3) and by Art. 90 (5), y'-b = ^(x^- a). (4) Combining (1) and the square of (4), (.-'—y{^^^=p'- (5) Combining (2) and (3), and the resulting equation with (4), ^ \ da' J da ^ ^ Dividing (6) by the square root of (5), and simplifying, ■y/da' + db^ = dp. Hence, if s represents the length of the evolute, by Art. 72, ds = dp', therefore As = Ap, 142 DIFFERENTIAL AND INTEGRAL CALCULUS. or in Fig. 24, P,P^ = r'P^ - P'P^, or the difference between any two radii of curvature of an involute is ^qual to the included arc of the evolute Art. 92. Mechanical Construction of an Involute from its E VOLUTE. From the two properties of the evolute established in Arts. 90 and 91, the involute may be readily constructed from its- evolute. Thus in Fig. 26, if one end of a string be fastened at N and the string be stretched along the curve NM having a pencil attached to the other Fig. 26. end, and then the string be gradually unwound from the evolute, always being in tension, the pencil will describe the involute MA. Every point in the string beyond TV- will describe an involute, as i? describes RS. So while any curve can have but one evolute, as NM is the only evolute of MA, it is evident that any curve may have an infinite number of involutes. A series of curves having the same evolute are called parallel curves. Art. 93. Envelopes of Curves. If in the equation of a plane curve of the form f(x, y, a) = 0, 7^ ENVELOPES. 143 different values be successively assigned to a, the several equations thus obtained will represent distinct curves, differing from each other in form and position, but belonging to the same class, or family of curves. Now, if a is supposed to vary by infinitesimal increments, any two adjacent curves of the series will, in general, intersect, and the inter- sections are points of the envelope. Hence, an envelope of a series of curves is the locus of the ultimate intersections of the consecutive curves. The quantity a, which remains constant in any one curve, is called the variable parameter. ?, _P2 P., Fig. 27. In Fig. 27, let AA\ BB', etc., represent curves of a series, and a„ 02, etc., their respective parameters ; then if ag — cii, as — ag, etc., diminish indefinitely, the ultimate intersections Pi, P2, P3, etc., will be points of the envelope. And, at the limit, the line Pj, Pg, joins two consecutive points on the envelope and on the curve BB', and hence is tangent to both the envelope and the curve BB', then the envelope is tangent to the curve BB'. Similarly, it may be shown that the envelope is tangent to any other curve of the series. Hence the envelope of a family of curves is tangent to each curve of the series. Art. 94. Equation of the Envelope of a Family of Curves. Let the equations of two curves of the series be f(x,y,a) = 0, (1) and f(x, y,a + Aa) = 0. (2) 144 DIFFERENTIAL AND INTEGRAL CALCULUS. The coordinates of the point of intersection of (1) and (2) will satisfy both (1) and (2), and hence will also satisfy fix, y,a^ Aa) -f(x, y, a) = 0, and f(x,y,a-\-Aa)-f(x,y,a) ^^ ,3. Aa ^ ^ As Aa approaches 0, the limit in equation (3) is dfjx, y, a) ^ ^ , , V da ' ^ ^ Now the coordinates of the point of intersection of two consecutive curves satisfy both (4) and (1). Therefore, by eliminating a between (1) and (4) the resulting equation is the equation of the locus of the ultimate intersections, which is the required equation of the envelope. For example, required the envelope of a series of curves repre- sented by y = ax-l^!^. (1) a being the variable parameter. ' Differentiating (1) with respect to a, x-^ = 0; (2) 2 hence ' a = — (3) X Combining (1) and (2), eliminating a, and reducing, y = l- which is the equation of the envelope 2/ = l-J (4) PROBLEMS. 1. Find the equation of the evolute of the ellipse AY -h B'x' = A'B\ (1) Here dy___mo . ^__^. PROBLEMS. 145 hence a = ^ -^^, and x = (^^— ^j ; ,^_{A^-B^f ^^, (_m\\ Substituting these values of x and y in (1), (Aa)^ 4- {Bbf^ = {A^ - JB^)^, which is the equation of the required evolute. 2. Find the equation of the evolute of the cycloid, x = r vers~^ ^ "" V2 ry — y\ Ans. a=r vers"' ( j + V— 2 r6 — 6'. 3. Find the equation of the evolute to the hypocycloid, x^ -\-y^ = A^. Ans. (a 4- b)^ +(a-b)^ = 2 A^. 4. Find the envelope of y^ -\-(x — of = 16, in which a is a variable parameter. Ans. y = ± 4:. 5. Find the envelope oi y = ax -\- —, a being the variable parameter. a Ans. y^ = 4: mx. 6. A straight line of given length slides down between rectangular axes ; required the envelope of the moving straight line. If c represents the length of the line and a and b the intercepts, the equation is M=^' , (^) the relation between a and b being a'+b' = (^. (2) Differentiating (1) and (2) with respect to a and &, gives ^EcUi = ^db, (3) a" b' ' ^ ^ and — ada = bdb. (4) 146 DIFFERENTIAL AND INTEGRAL CALCULUS. Dividing (3) by (4), X y X y a b a b 1 heace and X _y _ _ _ ^ ^~F' ^^' '^'~b'~¥TT'~?' a = (xc^^, b = (yc^^, which, substituted in (2), gives x^ -\-y^ = c^, which is the equation of the hypocycloid. Y Fig. 28. 7. Find the envelope of a series of concentric ellipses, the area and direction of axes being constant. A71S. If c = area, the equation of the envelope is xy = ± ——. 8. Find the envelope of a; cos a + 1/ sin a = p, in which a is the variable parameter. Ans. ^ -^y^ = p^. CHAPTER XIV. SINGULAR POINTS. Art. 95. Definitions. A singular point is a point of a curve which has some peculiarity not common to other points of the curve, and not depending on the position of the coordinate axes. The most important singular points are : 1st. Points of maximum and minimum ordinates; 2d. Points of inflection; 3d., Multiple points; 4th. Cusps; 5th. Conjugate points; 6th. Stop points ; 7th. Shooting points. Points of maximum and minimum ordinates have been considered in Chapter X., and points of inflection in Art. 81. Art. 96. Multiple Points. A multiple pfohit is a point common to two or more branches of a curve. There are two species of multiple points : 1st. Points of multi- ple intersection, or where two or more branchies of a curve intersect ; 2d. Points of osculation, or where two or more branches are tangent to each other. Multiple points are double, triple, etc., as two, three, or more branches meet at the same point. ■ At a multiple point there will be as many tangents, and therefore as many values of -^ as there are branches. If the values of — are dx dx unequal, the multiple point will be one of the first species, but if the values of -^ are equal, it will be one of the second species, (tx 147 148 DIFFERENTIAL AND INTEGRAL CALCULUS. Let u =f{x, y) = (1) be the equation of the curve freed of radicals. 'Then, by Art. 47, dy dx du dx du dy And since differentiation never introduces radicals when the func- tion contains none, the value of -^ cannot contain radicals, and there- dx ^ fore cannot have more than one value unless it assumes the form x' Hence the condition for a multiple point is ^ = - . Therefore, to examine for multiple points, — and — as obtained dx dy from the equation of the curve are placed equal to zero, and the corre- sponding values of « and y are found. If these values of x and y are real and satisfy (1), they may determine multiple points. Then -^= - (XX u is evaluated for the critical values of x and y, and every real value determines one branch passing through the multiple point. PROBLEMS. 1. Examine the curve y^ — {x — ayx = for a multiple point. du Here — = — 2 (a; — a) a; — (a; — a)^ = ; dx V / V / > and ^ = 2 y == 0. dy Solving (1) and (2) for x and y, gives , and (1) (2) x = a 2/ = -I But only the first point is to be examined, as the second point does not satisfy the equation of the curve. SINGULAR POINTS. 149 dy _ —2(x — a)x—(x — ay 3a; — a » dx" 2y ~ 2Vx = ± Va, when x = a. Therefore the multiple point is a double point of the first kind, as f shown in Fig. 29. Fig. 29. 2. Examine the curve x* -\-2 ax^y — ay^ = 0, for multiple points. dx — = 2ax'-Say' = 0. . dy (1) (2) Combining (1) and (2) gives three pairs of values for x and y, but the only pair that satisfies the equation of the curve is (0, 0). dy^ 4:a^-\-4.axy ^0 ^^en I ^ "" ^ dx 3ay^-2ax' O' Xy=0. dy dp Evaluating by Art. 59, and representing ^ by p and -f- by p\ cix dx dy _ _ 12 7^ -\- 4: ay -{- 4: axp dx 6 ayp — 4 aa; , {x = -, when < 0' (y = o, Sap , when ^x = and _ 24 a; + 8 aj9 4- 4 axp' _ 6 ap'^ -\- 6 ayp' — 4: a 6ap^ — 4 a Hence p(6 ajs^ — 4 a) = 8 op ; p = ^ = 0, +V2, and -V2. dx Therefore there is a triple point of the first kind at the origin. 150 DIFFERENTIAL AND INTEGRAL CALCULUS. 3. Examine y^ — aV — x^ for a multiple point. Ans. There is a double point of the first kind at the origin where ax 4. Show that the curve ]^ — x* -\-x^ has a point of osculation at the origin. Art. 97. Cusps. A (Msp is a point at which two branches of a curve are tangent to each other and terminate. Cusps are therefore multiple points of the second species. There are two kinds of cusps : 1st. When the two branches lie on -opposite sides of the common tangent; 2d. When the two branches are on the same side of the common tangent. Since a cusp is a particular kind of multiple point, curves are examined for cusps as for multiple points. But as a cusp is dis- tinguished from a multiple point by both branches stopping at the point, the curve must be traced in the vicinity of the point in question to determine a cusp. If the two values of J at the cusp have con- dor trary signs, the cusp is of the first kind, and if they have the same sign, the cusp is of the second species. The vertex of the semi-cubical parabola is a cusp of the first kind. [See point A, Fig. 25.] The curve (?/ — ^Y = ar^ has a cusp of the second species, determined as follows : Taking the square root of each meinber of the equation, y^x^±x^', (1) lience ^ = 2x±\x^, (2) ax and S=2±i^x^. (3) aar In (1), if X = 0, then 2/ = ; if x is negative, y is imaginary ; if x is positive, y has two real values. Hence, the curve has two branches on / SINGULAR POINTS. 151 the right of the T^axis which meet and terminate at the origin. The locus of the equation is shown in Fig. 30. In (2) ^ = 0, when a; = 0; hence the X-axis is tangent to both dx branches, and there is a cusp at the origin. In (3), when a value slightly greater than is substituted for cc, the two values of ^ are both positive ; hence the cusp is of the second species. t. Conjugate Points. Stop Points. Shooting Points. A conjugate or isolated point is a point whose coordinates satisfy the equation of a curve, but through which the curve does not pass. As the conjugate point is detached from the" curve, if the substitutions of a + 6 and a — h for x in the equation of the curve, h being very small, give imaginary values for y, then there is a conjugate point whose abscissa is a. Or, if at any point whose coordinates satisfy the equation of a curve, ^ is imaginary, this point will be a point through which no branches pass, and hence will be a conjugate point. For example, to examine y^ = (x — ly (x — 2) for conjugate points. The point (1, 0) will be such a point, for if some value a little greater or a little less than 1 be substituted for x in the equation, the 152 DIFFERENTIAL AND INTEGRAL CALCULUS. resulting value of y will be imaginary, yet the point (1, 0) satisfies the equation. Or by the second method : 3aj-5 • ^_ N"ow the point (1, 0) which satisfies the equation of the curve makes — imaginary, and hence is a conjugate point. In Fig. 31, JOT" is the curve and P is the conjugate point. A stop point is a point of a curve at which a branch suddenly ends. For example, to examine y = x log x for a stop point. Here, for any positive value of x, y has one real value ; when .t = 0, ?/ = ; when x is negative, y is imaginary ; therefore the origin is a stop point. Fig. 31. A shooti7ig point is a point of a curve at which two or more branches terminate without having a common tangent. For example, to examine y = x tan"^ - for shooting points. X Here, ^ = tan-'l- dx 1+3? When a; = 0, then v = 0. and ^ = ± -• dx 2 SINGULAR POINTS. 153 If X be positive and approach zero as its limit, ultimately ?/ = and -^ = ^; but if X be negative, ultimately ^ = and -^ = — ^. Hence (XX Ld (XX ^ two branches meet at the origin, one inclined tan W - j and the other inclined tan^( — -Y Therefore the origin is a shooting point. Stop points and shooting points occur only in transcendental curves, and may be discovered in any curve by tracing the curve in the vicin- ity of the singular points. • : CHAPTER XV. INTEGRATION OF RATIONAL FRACTIONS. Art. 99. Rational Fractions. A rational fraction is one whose numerator and denominator are rational. If the degree of the numerator is equal to or greater than the degree of the denominator, the fraction can be reduced by division to the sum of several integral terms and a fraction whose numerator is of a lower degree than its denominator. For example, ^' + ^ ^ dx^x'dx-^- xdx -3dx + , ^^ + ^ dx, x'-\-2x +1 a^-t-2a; + l in which the last term is the only fractional term. So it is necessary to consider only rational fractions in which the degree of the numera- tor is less than the degree of the denominator. A rational fraction is integrated by decomposing it into a number of simpler partial fractions, which can be integrated separately. Case 1. When the denominator can be resolved into n real and unequal factors of the first degree. f(x) Let ^^-^ dx represent a rational fraction, whose denominator may be resolved into the factors (a; — a), (a; — 6), ••• (a; — Q, real, unequal and of the first degree. Assume m = _A_+B^^...jL. (1) (jiix) x — a x — b X — c X — I in which A, B, O, ••• L are undetermined coefficients. Clearing (1) of fractions, f{x) = A(x -b)(x — c) ••• (x-l)-{- B(x - a){x-c)"'(x-l)-] + L(x-a){x- 6) ... (x - k). (2) Performing the indicated operations in (2) and equating the coeffi- cients of like powers of x in the two members by the Principle of 154 INTEGRATION OF RATIONAL FRACTIONS. 155 Undetermined Coefficients, will give n equations from which A^ B, C, etc., may be obtained. Or since (2) is true for all values of x, a may be substituted for x, which gives (a-b)(a-c)'-'(a-l) ^ ^ By substituting b for x, the value of B is obtained, and so on; finally when I is substituted for x, it follows that L = m (4) These values of A, B, C, etc., are substituted in (1), dx is intro- duced as a factor in each terra, and each term is then integrated. PROBLEMS. 1. Find rf + f-^^te. J XT -\- XT — (yx x^-\-x^-6x = x{x-\- S)(x - 2). Assume f + f-l ^A^^G^ a^-^x^-Qx X x-\-3 x-2 ^^ Therefore x^ + x-l = A{x + S)(x - 2) -]- Bx (x - 2) -{- Cx{x-\-3). Substituting x = 0, gives —1 = — 6A] hence A = ^. Substituting x = — 3, gives 5=155; hence B = ^, Substituting a? = 2, gives 5 = 10(7; hence C= J. Substituting these values of A, B and C in (1), introducing dx, and taking the integral of each member, dx rj^±x-i_. 1 rdx , r dx ,r = ^loga; + ilog(a; + 3) + ilog(a;-2) = log [a;^ (a; + 3)^ (a; -2)^]. f^^^^ = log[(x-inx-^2yi 156 DIFFERENTIAL AND INTEGRAL CALCULUS. J a' - b'x" 2 ah \a ~ hxj \ r {2x + ^)dx ^. (x-l)i ^ J ^ + x'-2x x^{x + 2Y ^ r(2 + ^X-^X^)dx , r ^/o . nI/o NT Case 2. When the denominator can be resolved into n real and equal factors of the first degree. Let ^^-Vi: dx represent a rational fraction whose denominator can be <^ {^) resolved into n factors each equal to a; — a. In this case the method of decomposition of the preceding case is not applicable. Take, for example, - — ^. {x — af Forming the partial fractions as before would give 2x'-{-x ^ A ^ B ^ C .^. {x — of X — ax — ax— a But if the fractions in the second member are added, 27?-\-x ^ A + B+C ^ {x — ay X — a ^ in which A + B -\- C must be regarded as a single constant, and evi- dently (2) cannot be an identical equation, as this would give three independent equations containing but one quantity, ^ + 5 + O, to be determined. The partial fractions are assumed as follows : /M^^!_+ -g + G ^..._A_. (1) <^ {x) {x - ay (x - a)"-^ (x - ay-^ (a; - a) ^ ^ Clearing (1) of fractions, f{x) = A + B{x-a)^C{x- ay+"-L{x-ay-\ (2) INTEGRATION OF RATIONAL FRACTIONS. 157 The values of A, B, C, etc., in (2), are found by the Principle of Undetermined Coefficients, then substituted in (1), after which dx is introduced, and each term is integrated separately. PROBLEMS. 1. Find C^^^^dx J (x — 1 r {x-iy Assume ^l+i ^ _A_ + ^^ + _CL_ . (x-if {x-if^{x-iy^(x-i) Hence x" -\-l = A + B(x-1) -\- C(x -ly = A+Bx-B^Cx^-2Cx+a Therefore 0=1, 5-2C = 0, and A-B^C=^1\ whence C = 1, 5=2, and yl = 2. Therefore (^^^^^dx= f-l^^ C^^^+ C^^ ^ r(^x^-2)dx ^12x^1^ J {x-\-2Y {X + 2Y ^ ^^ ^ ^ When the denominator of a rational fraction may be resolved into both equal and unequal factors of the first degree, the two methods must be combined. - . 4. r ^ = 5^±12_, w/x+iV J {x + 2y{x + 4y a^ + 6x + S \x + 2j r_^^zA^±I-dx = log [x(x- 3)^1 r dx ^ ^+JLiog^+^2 (a^-2y 4(05^-2) 8V2 ° x-^/2 158 DIFFERENTIAL AND INTEGRAL CALCULUS. Case 3. When some of the simple factors of the denominator are imaginary. As the denominator is real, the imaginary factors must occur in pairs, and of the forms •• X ±a-\- 6V— 1, and x ±a — bV— 1, whose product is the real quadratic factor (x ± af + b\ (1) For a single quadratic factor such as (1), the corresponding partial fraction will have the form ^t because each fraction of this {x ± a)- + b^ form increases by two the degree of the equation when it is cleared of fractions and therefore increases by two the number of the equations for determining A, B, C, .etc. ; hence its numerator should add two to the number of these undetermined constants. If the denominator contains n equal quadratic factors, being of the form [(x ± ay + b'Y, the partial fractions may be assumed as follows; f(x) ^ Ax -i- B Cx-{- D Kx-{-L ^2) (x) ~ \{x ± af + b'^Y ^ lix ± af + ?>--^]" '"' (a.- ± of ^b'' ^ ^ The values of A^ B, C, etc., are determined from (2), as in the pre- ceding cases. r 5^ J x^ + x" PROBLEMS. + x'-2 . x^ A , B , Cx-\-D Hence ^ = -i, B = \, 0=0, i> = i. / Mx ^ 1 r dx 1 r dx 2 r dx x^-^x'-2 qJ x + 1 6J x-1 sJ 0^ + : 11 X — 1 , V2 . -I X = I log \- -^— tan^ y) INTEGRATION OF RATIONAL FRACTIONS. 159 2- I 7 Tm TT = i 2,rc tan a; + log ^^ — ^ ^, • 3. f ^^^^ = logf^!±lf. B- / ^';7_^\'^ - ^ + log [(^ + 2)? (^ - 2)^]. . 6. r ^ =ilog^±£±l + J_arctan2^±i J{x' + l){^ + x + \) ^ ^ x' + l ^V3 V3 -: CHAPTER XVI. *- INTEGRATION OF IRRATIONAL FUNCTIONS. Art. 100. Irrational Functions. Very few irrational functions are integrable. When an irrational function cannot be directly integrated by one of the elementary formu- las, an effort is made to transform it into an equivalent rational func- tion of another variable by making suitable substitutions. When the rationalization can be effected, the integral may be found. Art. 101. Irrational Functions containing only Monomial Surds. An irrational function containing only monomial surds may be rationalized by substituting for the variable a new variable with an exponent equal to the least common multiple of the denominators of the fractional exponents in the function. —dx. Assume x = z^', then x^ = ^, x^ = z% and dx = 6 z'dz. Hence r x^ - 2 a J 1+Z^ . J 1 + ^2 = 6 Cfz' -2z'-z' + 2z'-\-z'-2z-l-}- ^^^-±l\dz = ^ - 2 z^ - ^z' -^ S:^ + 2 z' - 6 z^ - ez + 6 log(z^ + 1) -f- 6 arc tan z = ^-2x-^x^ -\-Sx^ -\-2x^-6x^ -6x^ + 6 log (x^ + 1) + 6 arc tan xK 160 INTEGRATION OF IRRATIONAL FUNCTIONS. 161 PROBLEMS. •^ x^ + a;3 a;^ ^ 3. r^£!^ = - isf^ + ^ + 1|! + 4«i + 16.J + 32 log (2 - xJ)l. 4. r^illda; = -4+^ + 21oga;-241og(a;T2 + l). ^ a:3 -(_ x^ ic^ a;T2 5. P^'^~^^' da; = 12(-|a;^ - f a;^ + 1^0;^^ _ 9^;^) + 1908 l^x^^ - f a;^ + 3a;^ - -^x^ + 81a;TiJ - 243 log (ojt'^ + 3)]. Art. 102. Functions containing only Binomial Surds of the First Degree. m A function which involves no surd except one of the form (a + bxy can be rationalized by assuming a -j-bx = 2;", as follows : Let /(a;, va + bx) be the function. Assume 2 = V« + bx ; then • z"" = a -\-bx, n^~Hz = b dx, dx = And x = b 2"— a Therefore Cf(x, ^/a + bx) cZa; = ^ Cff^l-^, z\ z^'-'dz, which is rational, and therefore can be integrated. M 162 DIFFERENTIAL AND INTEGRAL CALCULUS. PROBLEMS. xdx r *^ Vl+a; Assume 1 +x = z^\ then x = z^ — 1, and dx = 2zdz. Hence ("-4^ = r 2{z^-l)zdz ^ ^ C^,.^, _ ^,) = 12^-22 ^a? , _Vl + ic— 1 2. r ^ =iog */ JcVl + a; VI + a; + 1 3. C{x •{•■\/x-\-2+-^x-^2)dx = i(a; + 2)2- 2(05 + 2) +|(a; + 2)^4-f(aJ + 2)1 4. r vT+^a.' ^ ^^^ ^^ _^ V^TZl) + V^"=T. *^ V a; — 1 -^ (2r-2/)^ g r x^dx ^ 6a^ + 6a; + l ^ (4a; + 1)^ 12(4a; + l)^ 7. f , ^^ ^8[|(l+Vr3^)^-i(l+Vrr^)^]. •>' Vl+Vl-a; Assume 2 = v 1 + Vl — x. s^+^dx Art. 103. Functions having the Form -, in which n is (a + bx'y A Positive Integer. Expressions of this form may be integrated as in Art. 102. For example, find I — • Assume 1 — ar^ = 2^ ; then ar^ = 1 —z^, and xdx = — zdz. I INTEGRATION OF IRRATIONAL FUNCTIONS. 163 Therefore J^l^ = - f(lz:^ = -f(l -z^dz = -(^ - K) = i(l - '•O^ - (1 - =^)^- Art. 104. Functions having the Form fix, J^/ ^^H- \^^^ In this form the assumption may be made \ that then I ' and I therefore The substitution of these values will make the function rational. ' \cx + d' ^_ax-\-b cx-\-d' 02" - a a^-n(ad-bc)z-- 1 dz (cz^-ay PROBLEMS. Mx 3ar^ + 2 if-. '^ (l + x^)^ 3(1 +x2)^ •^ V2x2 + 1 3^ * -^ (2 + 3a;2^)^ 27(2 + 3ar2)t *• rT-^7^== = iiog(V3^^ + i) + }iog(V3:r^~3). •^ ar' + 2V3 — x^ 5 r 7 ^ ~^ da; _ _ 3 3 3 3 /I -a;' (l + a;)2 8V'V1 + 164 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 105. Functions containing only Trinomial Surds of the Form Va + 5a; -f car^. •■ Case 1. When c is positive. After factoring out Vc, the surd may be written V-A + Bx + ar*. Assume VA^i-Bx^i^ = z — x; then A-{-Bx = z'^-2zx, x = tjnA^ ' B + 2z and ^^^2(.-+^. + ^)c^., Therefore V^ + ^a; + ar^ = ^ - '^'~ ^ = ^' + ^^ + A jB + 22 2zH-5 Thus the given function may be transformed into an equivalent rational function of another variable. Case 2. When c is negative. After taking out the factor Vc, the surd may be written WA -\-Bx- x\ Assume a and /8 to be roots of the equation x^ — Bx — A = 0\ then Va^ — Bx — A = V(a; — a){x — /8), and V-4 + ^a; — ar^ = V(a; — a){fi — x). Let V(a; — «)(/? — a;) = (a; — a) 2; ; then (a; - a)(P - a;) = (a; - «)V, X «g^ + /8 2^ + 1' and ^^^2.(a-^)d.. Therefore V^ + 5a; - ar^ = (a; - a) z = (^~")^ r 4-1 Thus the given surd is expressed in rational terms of another vari- able. INTEGRATION OF IRRATIONAL FUNCTIONS. 165 PROBLEMS. 1. Find f ^ Bx + x" Substituting the values of dx and V-4 4- 5aj + ar^ as found in Case 1, gives r dx_ ^ ■\/A+B, _^ r2{z'' + Bz -\-A) dzx{2z + B) Bx-{-^ J iB + 2zyx(z' + Bz + A) If 5 = 0, (1) becomes = logfl + » + VZ+^T^I /: dx V^ + ar^ If A = ly (2) becomes = log[a; + V^ + a^]. •^^ Vl 4-ar 2. Find r ^^ ^ y/A + Bx — a? (1) (2) (8) Substituting the values of dx and wA + 5a; — a^ as found in Case 2, gives i ^A^Bx-^ ^ {f + Vf{fi-a)z Jl+z' I J a/s; c?a; '■/ V2 + 3x + a:2 die 2 arc cot f^^ — ^Y ^dx-G-x" \^-V 5. f— ^?=:=log(i + a^ + V^T^). = — 2 arc tan « = — 2 arc tan\/^ — -' ^ X— a = log[3 + 2a; + 2V2T3^+^]. 3 - x\^ 166 DIFFERENTIAL AND INTEGRAL CALCULUS. 6. I ^=1 = — - arc tan [ — ^ V 8. r^^= = '^logf^^±M^\ 1 /V^+W + a\ a \ X J or Art. 106. Binomial Differentials. Binomial differentials have the form af (a + hx'^ydXy in which m, n and p are any numbers, positive or negative, integral or fractional. 1st. If m and n are fractional, and the differential has the form « I ii^(a-{- hxydx, ^ = 2** may be substituted, and the expression becomes z'"(a + hz^yrtz'^-Mz = rtz"-^'''-\a + hz^ydz. 2d. If n is negative, and the differential has the form mf^ia 4- bx'^ydx, X = - may be substituted, and the expression becomes z x"^ (a + bx-'ydx = — 2;-"-2 (a + bz^dz. Hence, any binomial differential may be transformed into another, having integers for the exponents of the variable, and having a positive exponent for the variable within the parenthesis. In the following articles every binomial differential is assumed to have this reduced form. Art. 107. Conditions of Integrability of Binomial Differentials. As the exponent of the parenthesis is any number, let it be repre- aay be written p x"^ (a -\- bx'^ydx. (1) P sented by -, and the form may be written INTEGRATION OF IRRATIONAL FUNCTIONS. 167 1st. Assume (a + 6a;") = 2;* ; p then (a + 6af»)' = z^ (2) "M •-(•!^), (3) 1-1 and dx = -^z^-'l- — ^Y dz. (4) nh \ h J Substituting the values from (2), (3) and (4) in (1), p r^l ^ r" (a + bxydx = -^ z^-^'-' ■- — -] " dzj (6) nb V ^ / which is rational when — ^t_ is an integer or 0. 2d. Assume aa;~" + 6 = 2;«; (1) 1 1 then x = or{z!^ — h) % af' = a(:^-b)-\ (2) ar = a''{z^-b) ", (3) 1 _i_j and da; = -^a"(z«-6) " H'-^dz. (4) w Multiplying (2) by 6, adding a, and taking the ? power, (a4-6aJ")'=a'(^'-^) '2;^. (5) Taking the product of (3), (4) and (5), gives a;«(a + 6af')«da; = -£«'' « "(2? -6) ^" « ^z^+'-^da;, which is rational when ^ "^ + ^ is an integer or 0. Therefore, the n q binomial differential can be rationalized : 168 DIFFERENTIAL AND INTEGRAL CALCULUS. 1st. When the exponent of the variable without the parenthesis, in- creased by one, is exactly divisible by the exponent of the variable within the parenthesis. 2d. When this fraction increased by the exponent of the parenthesis is an integer. PROBLEMS. 1. Ym^iC^{2^^x^^dx. Here !!L±1_1 = ^_1 = 2; n 2, therefore the first condition of integrability is satisfied. Hence, let {2 + ^^ = z'^', then (2 + 3a^* = 2;, V - 2M =m' zdz and dx = 27 V7 5 3; = 2VR(24-3a;2)|_4(2 + 3a^f + |(2 + 3a^)^]. 2. r.-(i+^-^..=(i+4(|^!^. %/ oX 3. ^x^ia + b:^fdx = (a + b:^^{ ^ ^"f^'J ^^ » 4. f '^^ f2x + ^\l+:^rK INTEGRATION OF IRRATIONAL FUNCTIONS. 169 7. Ccc^ {a H- cc^ydx = ^\ (a + ic2)'»" _ s (^ + ar^^a + f (a + a^^^a^ PRACTICAL PROBLEM. A vessel in the form of a right circular cone is filled with water and placed with its axis vertical and vertex down. If the height = h, and radius of the base = r, how long will it require to empty itself through kn orifice in the vertex of the area a ? Neglecting the resistances, if the vessel is kept always full, the velocity of discharge through an orifice in the bottom is that due to a body falling from a height equal to the depth of the water. If v denotes the velocity and x the depth of the water, V = V2 gx. If dQ denotes the quantity discharged in the time dt through an orifice of the area a, dQ=adtV2gx. But in the time dt the surface whose area is S has descended the distance dXj thus dQ = Sdx. Hence Sdx = adt^2 gx^ or dt = — — a-\/2gx At the distance x from the vertex. ' K^ Therefore =x * irr'x'dx ^ 2 7r?-^Vfe dh?^2gx 5aV2g CHAPTER XVII. INTEGRATION BY PARTS, AND BY SUCCESSIVE REDUCTION. Art. 108. Integration by Parts. Integrating both members of d(uv) = udv -^vduy and transposing, j udv = uv — I V du. (1) Equation (1) is the formula for integration by parts. By this formula, I udv is made to depend on i v dii, and this new integral is frequently much simpler than the given one. PROBLEMS. 1. Find I a^logxdx. Let u = logo;, and dv = x^'dx; then c?w=— , and v= ^"''' X n-f 1 Substituting these values in the formula for integration by parts, J n + l n + lj n + lV n-\-lJ 2. I log a;da; = a;(logaj — 1). 3. Co sin fie = -eGose + sin 6. 170 INTEGRATION BY PARTS. 171 4. f!2g22££i^ = log ^. log (logx)- logo. 5 . fxe'^dx = €''(-— -\ J \a ay 6. I arc sin a; da; = a; arc sinaj + (1 — a*)*. 7. j a; cos a; da; = a; sin a; -f cos a;. xt3in^xdx = a; tan a; — — -h log cos a;. 10. f^e-dx = (a^-^-^ + ^-^-^\^. J \ a or a^J a >. Art. 109. Formulas op Reduction. When the integral, j x'^{a + bx^ydXy satisfies either of the conditions of integrability as given in Art. 107, it may be rationalized as explained in that article and then integrated. But by means of certain formulas of reduction, derived by the aid of the formula for integration by parts, the given expression may be made to depend upon simpler integrals of the same form. This method is called integration by successive reduction, and the integrals given by this method are generally in convenient form for integration between limits. 1. Formula A. f Assume j x'^{a -f- bx'^ydx = I udv = uv— Ivdu. (1) Let dv = x'^'Xa -f bxrydx, then u = af'-*'+\ Hence v = (<^ + ^^Y^ and du=(m-n-{-l) a;'»-'»c?a?. nb(p-\-iy 172 DIFFERENTIAL AND INTEGRAL CALCULUS. Substituting in (1), and putting a + 6x" = X, fx^X^dx = ^'" '''''^'^' _ ^ - ^ +^ fa;— "X^ifia;. (2) • Now faj'^-^XP+^da; = rx"'-"X^(a -f- bx'')dx = a pc'^-^X^dx + b Cx'^XPdx. (3) Substituting in the right member of (2) the value from (3), C^X'dx = ^-'^'X'^l _ (m - n + 1) a r^-„^,^^ (p + 1) n6(i> + l) m — w + 1 JV X^dx. Transposing the last term to the first member and solving for ■ x'^X^dx, J b(np-^m-^l) b(np + m-\-l)J ^ ' By Formula (^), the given integral is made to depend upon another of a similar form, having the exponent of x without the parenthesis diminished by the exponent of x within. 2. Formula B. = a Cx'^X^'^dx + b Cx'^^^'X^-'^dx. (1) Applying Formula {A) to the last term of (1), by substituting m + n for m, and p — \ for />, 6 fx-x-d^ = _^!!!^ _ «("» + !) f»"X'-'*., •/ np + m+1 np + m-\-\J which substituted in (1), by uniting similar terms, gives Cx^X'dx = ^"'''^'' + ^^ Cx^X^-Hx. (E) ; J np -\- 771 -{• 1 np -{- 7n -\- IJ - INTEGRATION BY SUCCESSIVE REDUCTION. 173 By Formula (B)j the given integral is made to depend upon another of a similar form, having the exponent of the parenthesis diminished by unity. Formulas (A) and (B) fail when np -h m + 1 = 0, but in this case ^?^^t — \-pz=Oj hence the method of integration of Art. 107 is appli- n cable. 3. Formula O. Solving Formula (A) for j x'^'^'X^dx, gives C^-X'd^ = f-"^"' - Hnp + m + l) r^^,^^ J a{m — n-\-l) a{m — n-^T)J Substituting — m for m — n, C^-«X'dx = ^"7'^'^;' + Km-n-np-l) T^-™.,^,^. (C) J — a(m — 1) — a(m — 1) J By Formula (C), the given integral is made to depend upon another of a similar form, having the exponent of x without the parenthesis increased by the exponent of x within. Formula (O) fails when m — 1 = ; in this case m = 1, and — m + 1 = ; hence the method of integration of Art. 107 is applicable. 4. Formula D. Solving Formula (J5) for j xT^X^-Hx, gives J anp anp J Substituting — p for p — 1, •/ an(p— 1) a?i(2> — 1) */ By Formula (Z>), the given integral is made to depend upon another of a similar form, having the exponent of the parenthesis increased by unity. Formula {D) fails when p — 1 = 0, but in this case the integral reduces to a fundamental form. 174 DIFFERENTIAL AND INTEGRAL CALCULUS. PROBLEMS. 1. Find f-^^. Here m = 3, n = 2, p = — ^, a= a^, and 6 = — 1. Substituting these values in (A), 3 «j 2. Find f-i^— . Here m = 0, w = 2, — p = — 3, a = 1, and 6 = 1. Substituting these values in (D), Applying (Z)) to the last term of (1), making mJtO, n=2, — p = — 2, a = 1, and 6 = 1, J(l + x^-'dx = ^(^ + '^"' + ij(l + x^r'dx = ;r7r^ — s: + i arc tan x. Therefore /(TT^a = 4irT^^ 8. Find r ^^ . f ^ = Cx'^a' - x^-kx PROBLEMS. 175 Here — m = — 3, n = 2, p = - J, a = (j?, and & = — 1. Substituting these values in (0), By Art. 105, Ex. 8, . a; _ arc sin — 2 a arc sm — Therefore r_^^^==- vV^+ 1 log^^^Vo^. 6. ("(1 - iB')^dic = Ja;(l - ar^^ + f a;(l - ic^)^ + f arc sina?. 7. J'VCa^ + a^(^^ = I V^^M^ + 1 log {X + V^M=^. -'vr^^ 1^5^5.3^5.3; 9. far^Cl - a;2)^c?a; = ^ a: (2 a;^ - 1)(1 - ar^^ + 1 arc sin a?. 10. fa^^ (1 _^ ^f^a. ^ (^^-2) (1 + a^)t. 12 ■ ^- /I " 13. f '^'i^ =.-(^g' + g^ + ^V2ax-r' + 4a^arcversg- -'V2ax-x^ V3 6 27 a Kbmabk. Reduce f-^^—^ to the form ^ x^ (^ a - xf^dx. J ^2 ax -01? -^ 176 DIFFERENTIAL AND INTEGRAL CALCULUS. ^ ^2ax-x' 2 a *- C x*dx x(S — xF) 15. I 7 = — ^ T — I- arc sm x. J (l-x')^ 2(1 -x")^ 9 16. r_^^ = _ 2 (3a;« + 4ar» + 8)Vir:^. CHAPTER XVIII. INTEGRATION OF TRANSCENDENTAL FUNCTIONS. INTEGRATION BY SERIES. Art. 110. Introduction. The method of integration by parts gives important reduction formulas for transcendental functions. Only a comparatively small number of logarithmic and exponential functions can be integrated by general methods. It is frequently necessary to resort to methods of approximation. Some of the principal integrable forms will be given in this chapter. Art. 111. Integration of the Form i f(x)(logxydx. It is assumed in this form that f(x) is an algebraic function and n is a positive integer. Let f(x)dx = dVf and (log a?)** = u ; then Cf(x)dx = v, and n (log xy-^ — = du. J X Substituting these values in \udv = uv— \v du, Cf{x) (log xydx = (log xyCfix) dx - C[n (log xy-^ — Cf(x) dx\ ; or, by making j f(x) dx = X, Cf{x)(iogxydx=X(iogxy - n C—(\ogxy-Hx. (1) Hence, whenever it is possible to integrate the factor f(x)dx, the given integral will depend upon another of a similar form, in which the exponent of the logarithm 'is diminished by unity. By repeated N 177 ITS DIFFERENTIAL AND INTEGRAL CALCULUS. applications of this formula the given integral will depend finally on the algebraic form | (x)dx. PROBLEMS. 1. Find Cxilogxfdx. Let xdx = dv, and (log xy = Wy oiy dx then — = '^) and 2 log x — = du. 2 x dx X Hence i a; (log xydx = — (log xy— I x^ log x ■ Similarly, | x log xdx=z— (log x)— i • J 2 J 2 X /Qiyt y3. a^ ' X (log xydx = — (log xy — — log X-] 2 . fx* (log ic)2 ^a; = ^ (log^ x-^logx-}- j\). rlogxdx xlogx , 1 /-, N 5. raf'(logx)^dai= ^[(logxy 2 loga; + -^1. c/ 71 H- 1 1_ 71 + 1 (n + 1)^J Art. 112. Integratiol'? of the Form | x"*a*^c?a;. In this form it is assumed that m is a positive integer. Let a'^dx = dv, and a;"* = w ; then -— = V, and mx'^'^dx = dw. w log a Therefore Cx'^a'^dx = _^!^ !??_ far^-ia'^'da;. */ nloga nloga*/ INTEGRATION OF TRANSCENDENTAL FUNCTIONS. 179 By successive applications of this formula, the exponent of x can be finally reduced to zero, and the given integral made to depend on the known form, I a'^'dx. PROBLEMS. 1. Find Cx^e'^'dx. Let e'^'^dx = dv, and a;^ = w ; then ( rcV'^dic = - e'^^'x^ ( xe'"dx. J a aJ Similarly, j xe'^'^dx = - e^^o; ( e'^^'dx. J a aJ Therefore Cx'e'^dx = — fx'' - ^' + ^\ J a \ a (Tj 2. ^xa'dx =J^(x — log a \ log a^ 3. Cx^e'dx =e'{x'^ -2x+2). 4. fa^e-da; == ~(:x? _ 5 a^ -f- 1 a; - -,\ J a\ a or (rj 5. r^ =_e-(ic2-|.2a; + 2). Art. 113. Integration of the For:.i i sin** ^ cos" ^ d^. '■/■ 1st. When either m or ti, or both, are odd positive integers. In this case the integration can be effected as in the following example : ^, fsin^ (9 cos-* (9 dd = C(l - cos^ 0) cos* sin dO = — i (cos^ 6 — cos*' 0) d cos cos^ cos^ 180 DIFFERENTIAL AND INTEGRAL CALCULUS. 2d. When m -\- n is an even negative integer. In this case the integration can be effected as in the following example : fsin^ e cos-^ edO= ftan^ cos"* dO = Aan^^sec^^d^ = Aan^ ^ (1 + tan^ 0) • d tan ^ = Atan^ ^ . d tan ^ 4- tan^ ^ . d tan 6) ^tan^ tan^ 3 5' 3d. When the form is not immediately integrable, neither of the aforesaid conditions being fulfilled. In this case the integral can be obtained by successive reductions. Let sin ^ = ic ; then sin"* 6 = x"", cos" ^ = (1 — af)\ and de = (1 - x'y^dx. Hence fsin"* cos" dO = fx"' (1 - af)~^dx. (1) \ Thus the given trigonometric form may be transformed into a binomial differential which may be integrated by means of the formulas of reduction. | For example, to find | sin* cos* dO. Let sin ^ = a; J then sin* = x\ cos* ^ = (1 — x^^^ and de = (l- x^y^dx. ^ Hence fsin*^ cos* dO = Cx\l - af)^dx. Applying Formula (A) twice, fx\l - x-)idx=- ^H^^ _ I . Jx(l -^i + i- if(l - ^^dx. (2) INTEGRATION OF TRANSCENDENTAL FUNCTIONS. 181 Applying Formula (B) twice to the last term of (2), Hence + I • i • i • i^(l - «^^^ + f • i • f . i arc sin a;. Therefore fsin^ cos* 6de = - ^—^ (sin^ ^ + i sin ^) + ^i^ sin 6 (cos^ ^ + f cos ^) PROBLEMS. cos^^ cos^^ . cos^^'^' .. fsin^^ cos^ ^d^ = - [ ^^^^ - -^^ + 6 4 ■ 10 2. f-^ = tan ^ + I tan^^ + i tan^^. J cos^ 3. f!H^!^ = sec ^ + 2 cos ^ - 4 cos^^. 4 r de ^ 1 _ 4cos^ _ 8 cos e J sin* cos^ ^ ~ cos ^ sin^ ^ 3 sin^ ^ 3 sin 6 5 . fsin^ cos^ edO=\ sin* ^ - | sin« d. g r sin'^^ ^^^ tan^(9 tan^^ J cos«^ 5 3* 7 . fsin* Ode = -\cosO (sin^ ^ + f sin ^) + 1 ^. 8. fcos* edO = ^ sine (cos^ ^ + | cos ^)- + | ^. « r "zi • 4^^/i sin ^ cos ^ /sin* ^ sin^^ 1\ , 6^ 11. rsec^ede = 52£l*22i + |log(secfl + tan«). J sm^ ^ 2 sm^ ^ 2 182 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 114. Iijtegration of the Forms j x"" cos (ax) dx and j X" sin (ax) dx. The formula for integration by parts is used, assuming that u = ic". Eviciently, each application of the formula will diminish the exponent n by one ; therefore, when n is a positive integer, the given form may be made to depend finally on j sin (ax) dx or j cos (ax) dx, each being a simple known form. For example, to find | x^ sin ic dx. Assume u = x^, and dv = sin x dx ; then du = 2x dx, and v = — cos x. Hence j a^ sin a;dx = — a^ cos « + 2 j .1; cos a;da;. Similarly, I x cos xdx = x sin x— i sin x dx = X sin X + cos X. Therefore 1 0? suixdx = — a? cos x -\- 2 xmix -\- 2 cos x. Art. 115. Integration of the Forms I 6*"= sin" a; da; and | e**cos"a;da;. Assume u = sin** x, and dv = e'^dx-, then du = n sin""^ x cos xdx, and v = — a Hence | e** sin** xdx = - e*^ sin** a; — - j e"'' sin**~^ x cos a; da;. (1) J a aJ Again, assume u = sin""^ x cos a;, and dv = e*"" da; ; then du = (n — 1) sin**"^ a; cos'^ x dx — sin** x dx = (n — 1) sin**"*^ xdx — n sin** a; da;, and v = — a INTEGRATION OF TRANSCENDENTAL FUNCTIONS. 183 Hence /ga* sinn-i ^ QQQ xdx= - e** sin"~^ x cos x ^^— I e"* sin"~^ x dx a a J + - fe** sin" a; da;. aJ Substituting in (1), and solving for I 6*"= sin" x da;, gives /._ . „ , e"' sin""^ X (a sin x — n cos x) e"'' sm" xdx = ^-— — ^ n^ + a" + ^'^f ""i^ fe"* sin"- 2 a; dx, (2) n^ + a^ J Each application of this formula diminishes the exponent of sin x by 2. By repeated applications n can be reduced to or 1, and the given integral will finally depend on I e^'dx = — , or j e'"' sin xdx. The value of j e*** sin a; da; is obtained directly from (2) by making In like manner J e** cos" 05 da; can be obtained. n = l. PROBLEMS. 1. j ar' cos a; da; = ar* sin a; + 2 a; cos a;— 2 sin a;. 2. I a^ sin a; da; == — x^ cos a; + 3a;2sina; + 6a; cos x — 6 sin x. e"* sin xdx = — (a sin a; — cos x). 4. j e* sin^ a; da; = — (sin^ a; + 3 cos^ a; + 3 sin a; — 6 cos a;). r are 2 7 6"'= COS a; (a cos a; + 2 sin x) 5. I 6°=" cos^ a; da; = ^^- -^ ^ J 4 + a2 2 e^ 184 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 116. Integration of the Forms j f{x) arc sin x dx, I f(x) arc cos ic dx, j f(x) arc tan x dx, etc. In these forms, f(x) is an algebraic function. Any one of these forms may be integrated by using the formula for integration by parts, assuming dv = f(x) dx. For example, to find /' ar* arc sinxdx. Assume dv = Mx, and u = arc sin x ; then 1? = — > and du = 3 Vn^ y? ■ , r Mx Hence | oc^ arc sin xdx = — arc sin x — ^ i - Vl^-^ Substituting a = 1, in Ex. 1, Art. 109, ^ Vl — ar Therefore j y? arc ^wvxdx = — arc sin a? + J (1 — ar^^(aj^ -}- 2). PROBLEMS. 1. I arc sin xdx = x arc sin aj -f-(l — a^)^. 2. I ^ — = X arc tan x — ^ (arc tan «)^— ^ log (1 -f a^. - /*a^ arc sina;daj ^ /-i , n\ r^ 2 • , a;^ , , 3. I — = — -^(3^ + 2) Vl — ar' arc sin a? -h — + |a;. */ Vl — ^ 9 4. la; arc cos xdx = ^ar^ arc cos a; — J a;(l — qi?Y + \ arc sin a?. f— ^?— = f- »/ a -f- 6 cos B ~ J I a[cos^g)+sin^(|)]f.[ 6cos^ dO COS'*! - 1— sm-* - INTEGRATION OF TRANSCENDENTAL FUNCTIONS. 185 dO -f (a + b)cos'(^) + {a-b)sm'(^ =/; sec-Y^W^ (a + &) + (a-6)taii2(| -/; ^6 d tan f - (a + 6) + (a-5)tan2^|^ ^ arc tanff ^^^y tan (^\], when a > 6. (1) li a ^3 ^ / / \ \ 3 C : c ) - t 4 X Fig. 32. Let y =^f(x) be the equation of the curve. And let OA = a, 0N= 6, and divide AN into n equal parts each denoted by Aa;, and erect ordi- nates at the points of division. Then, area of rectangle Pj-B =/(a)Aa;, area of rectangle P2C =f{Gb + Aic) Aa;, * Newton's Lemma XL, Principia^ Lib. I., § 1. 187 188 DIFFERENTIAL AND INTEGRAL CALCULUS. area of rectangle P^D =/(a -f2 Aic)Aa;, aifd area of rectangle P^_.iN=f(b — Aa;)Aa;. [jDheref ore, the sum of the n rectangles is f(a)Ax -\-f{a + Aa;) Aa; +/(a + 2 Aa^) Aa; + ... +/(5 - Aa;)Aa;, (1) which may be represented by ^ /(a;)Aa;, in which /(a;) Aa; represents each term of the series, x taking in succession the different values between a and h. Now as Aa; approaches zero, n increases indefinitely, and the limit of the sum of the rectangles is the required area AP^P^N. When Aa; becomes dx^ the symbol ^ is replaced by I , and the expression for the area, which is the sum of an infinite number of infinitely small rectangles, becomes C f{x)dxz=f{a)dx-^f{a-\-dx)dx+f{a-{-2dx)dX'-.-\-fQ) — dx)dx. (2) Assume I f(x) dx= (x) ; then / (x) dx = d {x) = (x ■{- dx) — (x). (3) Substituting successively for x in (3) the values, a, a -j- dx, a -\-2 dx, -•' b — dx, gives " ■"""-=/"(a)t?a; - ~(^(a-\- dx) — <}i(a), .---—. f(a-\- dx) dx = (a-{-2 dx) — <^ (a + dx), f(a + 2 dx)dx = cf> (a -{- 3 dx) — KJi (a -\- 2 dx), f(h-dx)dx =(b)-(b) — (a)y or r f(x) dx = {b)— (a). Therefore the area is found by integrating f(x)dx, substituting b and a successively in the integral, and subtracting the latter result from the former. INTEGRATION AS A SUMMATION. 189 — arc sm - 2 rjo PROBLEMS. 1. Find the area of the circle x^ -\-y^ = r^. Area of a quadrant = j f(x)dx= I ydx=.l (r^ — a^^dx. By Ex. 4, Art. 109, ~ 4 ' therefoi^e the area of the whole circle is ttt^. In order to obtain the area of the semi-segment OABC, Fig. 33, the superior limit of integration will be OA = X, and the inferior limit will be zero. Therefore- area OABC = C (r" - x^^dx ■^ — - — ^ + — arc sm -• 2 2 r Fig. 33. Evidently xjr'-x'y area of triangle OAB, and —arc sin - = area of sector OBC. 2 r 2. Find the area between the curve y^ ordinate through the focus. and X = 1. 4:X, the axis of X, and the Ans. A = ^. 3. Find the area of the ellipse a^ + 6V = a^b^. Ans. -n-ab. 4. Find the area of the hyperbola xy = 1 between the limits a; = a Ans. Log a. In this example it will be seen that the area of the hyperbola is the Naperian logarithm of the superior limit. For this reason Naperian logarithms are also called hyperbolic logarithms. 190 DIFFERENTIAL AND INTEGRAL CALCULUS. 5. Find the area of the cycloid x = rare vers - — \/2 ry — y\ dx= y^y . Here Therefore ■W^ry-y^ =^j: fdy Sttt^. W2ry-y^ 6. To find the area of y (l-\-x^) =^1 — o?, between the curve and the axes, in the first quadrant. The limits will be found to be a; = 1 and a; = 0. 1-0^ Therefore + 0^ dx Jo [_ or^ + lar' + lj r ^ HI = -- + ilog(a;2 + l)4-arctana; = .631972. 7. Find the area included between y^ = 2px and oi? = 2py, yI Fig. 34. The two parabolas intersect at (0, 0) and (2p, 2p) ; hence the limits of integration are 2p and 0. Area OBPA= f V2pxdx. Area OCPA =1 2p dx. INTEGRATION AS A SUMMATION. 191 Therefore area OBPC = TY V2^ -^\dx = 4^'. 8. Find tlie area included between y^ = 2x and y^ = 4:X — x^. Ans. 0.475. 9. Find the entire area within the hypocycloid x^ -{- y^ = a^. Ans. ^^ 8 10. What is the area of a theoretical indicator diagram when the steam is cut off at half-stroke, if the law of expansion is jj^y = 1 ? A71S. 1 + log 2. Art. 120. Areas of Plane Curves in Polar Coordinates. Referring to Fig. 35, it is required to find the area POP^, included between any plane curve AB and two vectors OP and OP^. B> Fiw. o5. Let the vectorial angles POX and PnOX be denoted respectively by y8 and a. If the coordinates of any point P be (r, 0), then the coordinates of Pj will be (r + Ar, 0+^0). The area of sector POS = i r • rAO = i r^AO. Then the sum of all the sectors POS, PiOSi, etc., may be repre- sented by ^ ^7-^ Ad\ and as A^ approaches zero, the limit of the sum of the sectors is the required area POP^, which will be given by the expression ^^ A = ^ \ rhW. 192 DIFFERENTIAL AND INTEGRAL CALCULUS. PROBLEMS. 1 . Find the area of the logarithmic spiral r = a^, between the limits ?'2 and r^. " Here dr = a» log a dO, and dO = ^^'^ ♦ rloga ' • Hence A=^ C-" rhW = -^— C\ dr = f-^T' Vri 2 log « Jr, |_4 log ajrj 4 log a 2. Find the area described by one revolution of the radius vector of the spiral of Archimedes ?' = aO. a j _ i T^" 22^^ _ 4:7rW — Vo ^ "" "■ ~3 3. Find the area of the lemniscate 1^ = o? cos 2 ^. ^?is. a^ 4. Find the area of a loop of the curve r = a cos 2 ^. ^?is. i tto?. 5. Find the entire area of the cardioid r = a(l — cos 6). q 2 2 6. Find the area of a loop of the curve r^ cos = a? sin 3 ^. ^ris. ^-^'log2. 4 2^ Art. 121. Rectification of Plane Curves referred to Rectangular Axes. By Art. 72, ' ' ds = [l +/'^YTda;, in which s represents the length of the arc. Therefore « =f [l + (|)]* cT. ; (1) the limits of integration being the limiting values of a?. The process of finding the length of an arc of a curve is called the rectification of the curve. If y be considered the independent variable, the formula is •-f['-(S)T* <»> in which d and c are the limiting values of y. -. . ■ AREAS AND LENGTHS OF PLANE CURVES. 193 If the arc PR, in Fig. 36, is to be rectified, the value of the first derivative is found from the equation of the curve, and substituted in Fia. 36. Formula (1) or (2). If Formula (1) is used, the limits b and a are OB and OA respectively ; if Formula (2) is used, the limits d and c are OD and OG respectively. ^ PROBLEMS. 1. Rectify the parabola '(f = 2px. Here dy p. dx 2/' ice dx=y^y. P Therefore s=Cfl = mV^ +1 log (2, + ^^Tf) + a (1) Here the value of the constant C may be determined by the first method of Art. 34. If the arc is estimated from the origin, then /S = when 2/ = 0, and these values substituted in (1) give O = |logi>+Ci hence Therefore O: P logp. yVfTf _^p 1 (y^-^p' + y\ 2p 2 \ p / 194 DIFFERENTIAL AND INTEGRAL CALCULUS. which is the length from the vertex to the point which has the ordi- nate y. Or if the limits of integration are known, for instance, if the length from the vertex to an extremity of the latus-rectum is required, then the limits are p and 0, and s = Vp' + y\ E log (y + Vp' + /)T = ip V2 +g log (1 + V2). 2 2p 2 2. * Rectify the semi-cubical parabola y^ = aa^. Ans. -I-A +|aa;Y--^. 27 a\ ^ J 21a 3. Rectify the curve whose equation is y"^ = —y and determine the length of the curve from the origin to the point whose ordinate is 10. Ans. 19.0248. 4. Rectify the circle ar^ -j- 2/^ = i-^. Here g = 4 TYl 4-^^(^a;=:4r T ^^ =27r7\ Jo \^ 2/7 ^^ vV - x" But as the result is in circular measure, the circle is a non-rectifiable curve. An approximate result may be obtained by a series. A r d^ A [^ . ^ , 1 • 3 x^ , "!'■ '='U ^j^^ = ^'\j^2:^?^2riTw?^'''i L ^2.3^2.4.5^2.4.6.7^ J Therefore L 2.3 2.4.5 2.4.6-7 J From this equation the approximate- value of ir can be determined with any required degree of accuracy by taking a sufficient number of terms. 5. Rectify the ellipse 2/^ =(1 — e^){a? — x^. Here ^ = _ (1 _ ,^) ? = _ ^^^Izi^^. dx y Va^-ar^ * The semi-cubical parabola was the first curve whose rectification wa> effected algebraically. (Neil, in 1657, Phil. Trans., 1673.) AREAS AND LENGTHS OF PLANE CURVES. 195 Hence s = 4 CJ^Lf^dx = 4.r —^ — (a^ - e'x^^ Jo M a^- x^ Jo Va^ - x^ J« V^^^^V 2a 2.4a3 '" J 6. Rectify the hypocycloid x^ -}-y^ = a^. Ans. aS = I a^x^ ; the entire curve = 6 a. 7. Rectify the cycloid x = r arc vers - — V2 ry — y^. Here ^= ^ • ^2/ V2ry-y^ Therefore s = 2 f Y^r^^ V di/ = 8 r. Jo \2r-yJ Art. 122. Rectification of Curves in Polar Coordinates. By Art. 73 (2), -[-HI)']'*- Ti,,...„ -r['*+(i)T'"- PROBLEMS. 1. To find the length of the cardioid r = a (1 + cos 0). Here ^ = -asin^. dO Therefore ^ = ^ f" f ^' ^^ + ^°^ ^)' + ^' ^^^' ^]^ ^^ = 2a C\2-j-2cose)^de = 4:a CcosidS c/o c/0 2 = 8 a sin - = 8 a. 196 DIFFERENTIAL AND INTEGRAL CALCULUS. 2. Eectify the spiral of Archimedes r = aO. 2a ^2 ^\ a J 3. Rectify the logarithmic spiral logr = ^ between the limits Vi and r-o. Ans. (1 -f- m^)^(ri — Vo). 4. Rectify the curve r = a sin' Ans. Art. 123. The Common Catenary. The common catenary is the curve assumed by a flexible cord of uniform thickness and density, fastened at two points, hanging freely and acted upon only by the force of gravity. As the cord is regarded as perfectly flexible, the only force acting at any point of the cord is a pull in the direction of the cord at that point, which is called tension and is a function of the coordinates of that point. Y \ ^ / v /' N / X" 0' Fia. 37. In Fig. 37, let 0, the lowest point of the curve, be the origin, and let the horizontal line through be the X-axis, and the vertical OY be the F-axis. Let (x, y) be any point P on the curve, s the length of OP, and c the length of the cord whose weight is equal to the tension at O. If the weight of the unit of length be taken as the unit of weight, the length s will represent the weight of the arc OPf and the length c will represent the tension at 0. AREAS AND LENGTHS OF PLANE CURVES. 197 Then the arc OP may be regarded as a rigid body in equilibrium under three forces : the tension at P in the direction of the tangent, the horizontal tension c at the origin, and the weight s acting verti- cally downward. Draw PN tangent to the curve, and PS parallel to OX at P. Then by the triangle of forces, the sides of the triangle PSN will represent the three forces acting on the arc OP. Therefore |g ^ weight of OP ; SP tension at O c hence -^ = -. (1) dx c Differentiating (1), substituting the value of ds, and reducing, \/'+v* dx dy\- c (2) Integrating (2) and noticing that when a; = 0, -^ = 0, dx '-[lW'H2)]=v or . », . I dx ^ \dx whence -^=i(e<= — e *). dx ^^ ' • Integrating (3), and noticing that a; = when y = 0, (3) y = 'L{^ + e'')-c, (4) which is the required equation. Removing the origin to the point 0', which is at a distance c below 0, the equation becomes 3, = |(e^ + e')- («) 198 DIFFERENTIAL AND INTEGRAL CALCULUS. In order to rectify the catenary, (5) is differentiated, giving | = i(J_e-j), asm(3), from which is obtained Therefore s = ^ C\e^ + e~'']dx = |(e' - e~}. PROBLEM. What is the curve in which the cables of a suspension bridge hang ? Ans. A parabola. CHAPTER XX. SURFACES AND VOLUMES OF SOLIDS. Art. 124. Surfaces and Volumes of Solids of Revolution. 1st. Surfaces. In Fig. 38, let the plane curve MN revolve about the X-axis. Let M be a fixed point and P any other point of the curve whose coordinates are {x, y). / < 5 A ^ -R ^N C A B Fig. 38. Assume MP — s and PQ = As, then the coordinates of Q are (x -\- Ax, y + Ay). Let S represent the area of the surface generated by the revolution of MP, and AS the surface generated by PQ. Draw PR and QT, each equal in length to As, and parallel to OX. In the revolution PR generates the convex surface of a cylinder whose area is 2 Try As, and QT generates the convex surface of a cylinder whose area is 2 tt (2/ -f Ay) As. Obviously the area of the surface generated by PQ lies between the areas of the cylindrical surfaces. Hence 2 7r2/ As < AaS < 2 77 (2/ H- Ay) As. Therefore, as As approaches zero. dS = 2iry ds, 199 200 DIFFERENTIAL AND INTEGRAL CALCULUS. and S=C2 7ryds (1) dx. (2) = 2./,[ 1 + '^'^ dx ^ In like manner for the surface generated by revolving the curve about the F-axis, S = 2 wCx ds = 2 ttCx ["l + (^)'f ^y- (3) The surface of a zone, included between two planes perpendicular to the X-axis and corresponding to the abscissas 6 and a, is = 2 7rjjds. . (4) S 2d. Volumes. Let V denote the volume generated by the surface in- cluded by the curve MP, the ordinates ML and PA, and the X-axis. Let A V represent the volume generated by APQB. The volume of the cylinder generated by APGB is iry^AXf and the volume of the cylinder generated by ASQB is ir(2/ -f- AyY^x. Obviously, iry^Ax < A F< tt (?/ + /^yyAx. Therefore, as Aa; approaches zero, dV= Try^dx, and V= TT I y^dx. (5) In like manner, for the volume generated by revolving the curve about the F-axis, V=7r Ca^dy. ' (6) PROBLEMS. I 1. Find the surface of the sphere generated by revolving the circle x^ -{-y^ =z r^ about a diameter. J Here y=(7^-a^)^ and — = --• (XX y SURFACES AND VOLUMES OF SOLIDS. 201 -Therefore S=2Tr ry(l-{-^^ dx = 2'jrr C dx^ 4.-^7^. 2. Find the volume generated by revolving the parabola y^ — 2px about the X-axis. V= TT i 2pxdx =pTrx^ = 1 tt/ • X. 3. Find the volume of the cone generated by revolving y = x tan a, when a is the semi-vertical angle of the cone. Ans. V=\ volume of circumscribing cylinder. 4. Find the volume of the sphere generated by revolving oi^-{-y-=r^ about the X-axis, and also the volume of a spherical segment between two parallel planes at distances b and a from the centre. Ans. J TTT^ and tt [r2(6 - a) - ^ (6» - c^)]. 5. Find the surface and volume of the prolate spheroid generated by revolving y^ = (1 — e^{a? — x') about the X-axis. Ans. S = 27r¥ -] arc sin e and V — — - — e 3 6. Find the surface and volume of the right circular cone, gener- ated by revolving the line joining the origin with the point (a, b) about the X-axis. / ^ , , n , ^r 'rrab^ Ans. S = 7rb Va* -h b^ and V= ——. S 7. Find the surface generated by the cycloid y = r arc vers - -f- V 2 rx — oc^, r when it revolves about its axis. * Ans. 8 irr^iv — f ). 8. Find the volume generated by the cycloid a; = r arc vers - — V2 ry — y\ T when it revolves about its base. Ans. 5 ttV. 9. Find the surface and volume of the annular torus, generated by revolving the circle a? -\- {y — 5"y = 4", about the X-axis. Ans. S = 394.79 sq. in., and V= 394.79 cu. in. 202 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 125. Surfaces by Double Integration. In Fig. 39, let {x, y, z) and (x -{- dx, y -\- dy^ z + dz) be the coordinates of two consecutive points P and E on the given surface whose equation is known. Through P and E pass planes parallel to the planes XZ and *YZ. These planes will intercept an element PE of the curved surface, which is projected on the XY^plane in BS = dxdy. Fig. 39. Let S represent the required area, and dS the area of the element PE. The area of PS is evidently equal to the area of PEj multiplied by the cosine of the angle which PE makes with XY. Representing this angle by y. hence area PE • cos y = dxdy^ area PE = dxdy • sec y. By the aid of analytical geometry of three dimensions,* in which — and — are partial derivatives from the equation of the dx dy ^ ^ given surface. * See Appendix ; Note A. SURFACES AND VOLUMES OF SOLIDS. 203 Therefore area FE = dS = fl -\-f^\-^f^YYdxdyy The effect of the 2/-integration, x remaining constant, will be to give the sum of all the elements similar to PE from W to M: hence the limits of the ^-integration will he y = CM=^^ OM'^—xr and y = 0. The effect of the subsequent a;-integration will be to give the sum of all the strips similar to WMN forming the given surface ; hence the limits of the second integration are x = OA and a; = 0. Art. 126. Volumes by Triple Integration. The given volume is supposed to be divided into elementary rectangular parallelopipeds by planes parallel to the three coordinate planes ; such an element of volume is represented by kl in Fig. 39. The volume of such an elementary parallelopiped is dx dy dz ; hence the whole volume is F= C C Cdxdydz. (1) The limits of integration are obtained from the equation of the bounding surface, being so chosen as to embrace the entire volume. If the volume included between the three coordinate planes and the curved surface is required, the limits are found as follows : The effect of the ^-integration is to sum all the elemental parallelo- pipeds in the prism PS-, hence the limits of the first integration are PR = z =f(x, y) and 2 = 0. The effect of the ^/-integration is to sum all the elemental prisms in the slice WN\ hence the limits for the second integration are CM=y=f(x) and y = 0. The effect of the a;-integration is to sum all the elemental slices composing the whole volume ; hence the limits of the third integration are x — OA and x = 0. Ou ?y z For example, to find the voli^ne of the ellipsoid — -f ^ + — = 1, cut or Ir (f off by the coordinate planes. - Here the limits of the 2^-integration are c a/1 i~ 2 ^^^ ^? *^® 204 DIFFERENTIAL AND INTEGRAL CALCULUS. limits of the y-integration are 6 -vl ^ and 0, and the limits of the ic-integration are a and 0. "^2 ~2 j ^ 1 ;; — r:;, and yi = b\l in the formula, a^ b^ ^ a? gives for the whole ellipsoid 9 /•a r»y\ f*tx F= 8 j 1 \ dx dy dz. Integrating on the hypothesis that z is the only variable, Integrating again, now on the hypothesis that y is the only variable, Integrating finally with respect to x, gives V='^^r{a^-x^dx=^^abc, PROBLEMS. 1. Find the surface of the sphere qi? -\- y^ -\- z^ =. a*. Here |2= _2, and |^ = -?!. dx z oy z dxdy Therefore 5= JJ(l + J + J)^.x.,=//_^ a^-/ Integrating with respect to i/, between y = Va^ — 3? and y = 0, SURFACES AND VOLUMES OF SOLIDS. 205 Integrating with respect to «, between x = a and a; = 0, Jo 2 2 which is the area of one-eighth of the surface of the sphere. 2. A sphere x^ -\- y^ -j- z^ = a~ is cut by a right circular cylinder y- = ax — x^. Find the area of the surface of the sphere intercepted by the cylinder. Ans. 2a^(7r — 2). 3. Find the surface intercepted by two right circular cylinders a^-\-z^= a^ and x' -{- y^ = a^. Ans. 8 al 4. Find the volume of a right elliptic cylinder whose axis coin- cides with the X-axis and whose altitude = 2 a, the equation of the base being c^ + b^z'^ = b^c-. Ans. 2 irabc. 5. Find the volume of the solid contained between the paraboloid of revolution a^-\-y^ = 2zj the cylinder ar^ + y^ = 4 a;, and the plane z = 0. xi+yi Ans. 2pP'''"'f ' dx clydz^ 37. 699- «/0 »/0 t/0 6. Find the volume of the solid cut from the cylinder a^-\-y^ = a^ by the planes z = and z = x tan a. Ans. f a^ tan a. 7. Find the entire volume bounded by the surface x"^ -\- y^ -\- z^ =:-s/2E. Ans. 44.88. CHAPTER XXI. CENTRE OF MASS. MOMENT OF INERTIA. PROPERTIES OF GULDIN. Art. 127. Definitions. The definitions of this article are taken from Mechanics and are here assumed without investigation. The moment of any force with respect to an axis perpendicular to its line of direction is the product of the magnitude of the force by the perpendicular distance from its line of direction to the axis. The moment of a force with respect to a plane parallel to its line of direc- tion is the product of the force by the perpendicular distance from its line of direction to the plane. The force exerted by gravity on any body is proportional to the mass of the body, and hence the mass of the body may be taken as the measure of the force exerted on it by gravity. The centre of mass of a body is that point so situated that the force of gravity produces no tendency in the body to rotate about any line passing through the point ; hence it may be regarded as the point at which the whole weight of the body acts. The centre of mass is sometimes called centre of gravity and centre of inertia. The moment of inertia of a body with reference to a straight line, or plane, is the sum of the products obtained by multiplying the mass of each element of the body by the square of its distance from the line or plane. Points, lines and surfaces, as here considered, are supposed to be material bodies. Lines, surfaces and solids are regarded as being com- posed of an infinitely large number of indefinitely small particles. The weight of a body is the resultant of the weights of all of its elemental particles acting in vertical lines, and the resultant of this system of parallel forces passes through the centre of mass. 206 CENTRE OF MASS. 207 Art. 128. General Formulas for Centre of Mass. Assume a system of rectangular coordinate axes, retaining a fixed position with reference to the body, the plane XY being horizontal. Let a small particle of mass at any point (x, y, z) be represented by Am. Then the force exerted by gravity on Am is measured by Am in a direction parallel to the Z-axis. Fm. 40. ^f the mass of Am were concentrated at the point (x, y, z), the moment of the force exerted on Am with respect to the plane YZ would be icAm ; and the sum of the moments of all the elements of the body with reference to this plane would be 2a;Am. The resultant force of gravity is ^SAwi, and if the coordinates of the centre of mass be represented by (x, y, z), as the centre of mass is the point through which the resultant passes, S^Am will be the moment of the resultant with reference to the plane YZ. But by the principle of moments, the moment of the resultant of any number of forces is equal to the algebraic sum of the moments of the forces. Hence, x^\m = 2a;Am. If now Am diminishes indefinitely, X I dm = I xdm. I xdm Therefore x='^ . I dm (1) 208 DIFFERENTIAL AND INTEGRAL CALCULUS. I ydm Similarly, ^^^r ' ^^^ j dm /zdm .-— (3) I dm The mass of any homogeneous body is the product of its volume by its density. If k represents the constant density and dv the ele- ment of volume, then kdv = dm, and (1), (2) and (3) become I xdv ^=^. (4) iydv I zdv and ^='V-' (^) fdv If the body is a material line in the form of the arc of any curve, and if ds is the length of an element of the curve. Formulas (4), (5) and (6) become I xds ^=^' (7) fas Cyda y=J—~, (8) zds fas If the curve is a plane curve, it may be taken in the plane XT, in which case z will be zero. CENTRE OF MASS. 209 PROBLEMS. 1. Find the centre of mass of an arc of a circle, taking the diam- eter bisecting the arc as the X-axis and the left vertex as the origin. In Fig. 41, let AOB be the arc. Fia. 41. The equation of the circle is ^ = 2 aa; — sc*; hence dy _ (a — x)dx V2 ax — sc^ ds = Vdoc^ H- dy^ = adx f^^^ a r' Therefore x=^ = - I Cds ^^'^ V2aa;-ar^ « V2 ax — x^ xdx =«(_V2^^3^ + s)=a-52^. 2. Find the centre of mass of an arc of the hypocycloid x^ -\-y^ = aJ, between two successive cusps : Here dy = -(^'^dxi hence ds = -Vda^ H- dy^ = y^^x'^ -{-y ^dx = V tt* - x^ V--* + -T^ ■dx = (-]^dx. a^ — x^ 210 DIFFERENTIAL AND INTEGRAL CALCULUS. Therefore Similarly, OM==x = SHtf" Jo \xj dx = i' Fig. 42. fa. 3. Find the centre of mass of the arc of a semi-cycloid. Ans. x= (Tr — ^)a, y = Art. 129. Centre of Mass of Plane Surfaces.* If rectangular coordinates are used, dv = dA — dxdy, and Formulas (4) and (5) of Art. 128 become I I xdxdy \ { dxdy \ \ ydxdy y=itLtL I I dxdy The centre of mass of a plane area is sometimes called the centroid of the and (1) (2) CENTRE OF MASS. 211 PROBLEMS. 1. Eind the centre of mass of the area included between the parab- ola if- = 2px and the double ordinate whose abscissa is a. I xdxdy I xydx . __ J-y =z^ I dxdy I ydx J~y Jo r*a 3 I ^2j)X'-i _./0 ■dx x^dx I a. 2. Find the centre of mass of the semicircle y?-\-y^ = r^ on the right of the F-axis. I I xdxdy I xydx ( ( "dxdy i 'ydx Ja J y Jo T^dx I V?" — x^dx 4r 3. Find the centre of mass of an elliptic quadrant whose equation b IS a . - 4a - 46 Ans. X = — , y = - — 3 TT O TT 4. In Fig. 43, ABD is a segment of a parabola cut off by an ordi- nate, and BE is parallel to Ax. 1st. Determine the distance of the centre of mass of ABD from Ax. . 3y 8 2d. Determine the centre of mass of ABE. , (Zx 3y 5. Find the centre of mass of the cycloid. Ans. X = Trr, 2/ = f r. 212 DIFFERENTIAL AND INTEGRAL CALCULUS. 6. Find the centre of mass between ?/"* = a;" and ?/" = x"". Ans. x = y = (^+^y (m + 2n)(2m + n) ,Art. 130. 'Centre of Mass of Surfaces of Revolution. If a curve in a plane with the X-axis be revolved about this axis, then dv = 2'iryds\ hence, by Art. 128 (4), I 2irxyds I xyds i *2-jryds iy ds PROBLEMS. 1. Find the centre of mass of the convex surface of a right cone, generated by the line y = ax. . Here ds = ^dx^ + dy^ = Vci^ -r ^dx; I ax^ Vci^ -i-ldx hence '^ = ~^ ZZHZT — = i^- I ax Va^ -{- Idx 2. Find the centre of mass of the surface generated by the revolu- 'fj tion of a semi-cycloid x = a vers~^ - — V2 ay — y^ about its base. Here ds = __ V2ady V2 a — y p xydy hence x= ^^^ =||a. 3. Find the centre of mass of the convex surface of a hemisphere whose radius is equal to 10. Ans. x = 5. CENTRE OF MASS. 213 Art. 131. Centre of Mass of Solids of Eevolution. If a solid be generated by the revolution of a plane curve about the X-axis, then dv = 2irydydx\ hence, by Art. 128 (4), I j xydxdy jjydxdy PROBLEMS. 1. Find the centre of mass of a right circular cone, whose convex surface is generated by revolving y = ax about the X-axis. •*' /*x /•ax J. X """ dx —dx 2. Find the centre of mass of a paraboloid generated by ^ = 4: ax. n'*' xydxdy - x = ' ff'^'ydxdy J I 2aMx = |aj. J-»x <* 2axdx 3. Find the centre of mass of a hemispheroid generated by 52 y^ = — (2ax — x^. Ans. -fa. CL 214 DIFFERENTIAL AND INTEGRAL CALCULUS. Art. 132. Moments of Inertia of Surfaces. In Fig. 44, the curves AB and CD and the ordinates LN and PM intercept a plane surface PLSR, whose moment of inertia is required. The surface is supposed to be divided into rectangular elements by lines parallel to the coordinate axes. f D J ) - r L k-^^ c^^^^ " ~ -.^^^^^ n ^^^■^B f J J R W V ^ ^^--0 Fig. 44. Let (a;, y) be the coordinates of any point as /, then (x -{- dx, y + dy) will be the coordinates of g^ and dxdy will be the area of the element^. The moment of fg about X= y^dxdy. Let the equations of AB and CD be ?/ = f{x) and y=(x) respec- tively ; and let 0N= b and OM—a. If X be regarded as constant, while y varies from <^ (x) to /(a;), the integration will give the moment of the vertical strip WQTV. Then in the second integration, x varying from a to h, the sum of the moments of all the stripe composing the area PRSL will be given. Representing the moment of inertia by M. I., fdxdy PROBLEMS. 1. Find the moment of inertia of a circle about its diameter. M.I. = ( I '^ *^ yHxdy */-rt/-v'r2-i2 PROPERTIES OF GULDIN. 215 2. Find the moment of inertia of a rectangle about an axis through its centre parallel to one of its sides. Let 2 h and 2 d denote the width and length respectively, the axis being parallel to h ; then M. I. = C Cy^dx dy = ^ hd\ 3. Find the moment of inertia of an isosceles triangle about an axis which passes through its vertex and bisects its base. Let a = the altitude and 2 6 = base, and take the origin at the vertex and the axis of moments as the X-axis ; then -X Art. 133. Guldin's Theorems.* I. Let a plane curve in the same plane with the X-axis revolve about the X-axis. The ordinate of the centre of mass is ^Jyds -, by Art. 128 (8). Therefore 2'jry ' s — 2ir {yds. (1) But by Art. 124 (1), the second member of (1) is the area of the surface generated by the revolution of the curve whose length is s about the X-axis, and the first member is the circumference described by the centre of mass, multiplied by the length of the curve s. Hence, if a plane curve revolve about an axis in its own plane external to itself, the area of the surface generated is equal to the length of the revolving curve, multiplied by the circumference de- scribed by its centre of mass. * Sometimes called Theorems of Pappus, as they were first stated by Pappus. 216 DIFFERENTIAL AND INTEGRAL CALCULUS. II. A plane area revolves about the X-axis. The ordinate of the centre of mass of the plane surface is •-1 11 ijdxdy y=-Y-r ' by Art. 129 (2). •; ijdxdy Therefore ^"ry \ I dxdy = 27r I I ydxdy =:7r i yHx. (2) But by Art. 124 (5), the last member of (2) is the volume gener- ated by the revolution of the area ; and in the first member, I i dxdy is the revolving area. Hence, if a plane area revolve about an axis external to itself, the volume generated is equal to the area of the revolving figure, multiplied by the circumference described by its centre of mass. If the curve or area revolve through any angle instead of making an entire revolution, 6 must be substituted for 27r in equations (1) And (2). PROBLEMS. 1. Find the surface and volume of the ring generated by revolving a circle whose radius = r, about an external axis distant & from the centre of the circle. Aii&. S = ^ n^ah^ V= 2 7rV6. 2- Find the volume generated by an ellipse revolved about an axis distant 10 from the centre ; the semi-axes being 10 and 5. Ans. 9869.6+. 3. Find the surface and volume generated by revolving a cycloid about its base. Ans. S = -^Tra', V= 5v^a\ CHAPTER XXII. DIFFERENTIAL EQUATIONS. Art. 134. Definitions. A differential equation between two variables x and y is an equation containing one or both of the variables x and y and one or more deriva- tives, such as -^, — ^, —^, etc. dx dxr dor The order of a differential equation is that of the highest derivative which it contains. The degree of a differential equation is that of the highest power to which the highest derivative which it contains is raised. The solution of a differential equation consists in finding a relation between x and y and constants, from which the given equation may be derived by differentiation; this relation is called the primitive. The solution requires one or more integrations, and each integration intro- duces an arbitrary constant ; hence the solution of a differential equa- tion of the nth order will give an equation containing n arbitrary constants. The same primitive may have several differential equations of the same order. For example, given the equation ay -\- bx -\- c = 0. (1) By differentiating, a^i-b = 0. (2) dx Eliminating a between (1) and (2), bx^ + cf- dx dx Eliminating b between (1) and (2) 2,a;^-hc^-62/ = 0. (3) dx dx ay-^c-ax^ = 0. (4) dx 217 218 DIFFERENTIAL AND INTEGRAL CALCULUS. In this example, equation (1) is called the complete primitive, and equations (2), (3) and (4) are differential equations showing the same relation between the variables. A?iT. 135. Differential Equations of the First Order and Degree. The general form of the differential equation of the first order and degree is Mdx + Ndy = 0, (1) in which Jtf and ^ are functions of x and y. This equation may be put in the form ax The most obvious method of solving a differential equation of the first order and degree is by means of the separation of the variables, whenever practicable. The variables are separated when the coefficient of dx contains the variable x only, and the coefficient of dy contains the variable y only ; that is, when the equation can be reduced to the form Xdx 4- Ydy = 0, in which X is a function of x only, and r" is a function of y only. Let the form be XYdx-\-X'Tdy=^0, (2) in which X and X are functions of x only, and T and F' are func- tions of y only. Dividing by XT, fF + |c - y = vx, and -^ = x \-v, <<> ^^ dx dx dv , /./ N in which the variables can be separated, giving dx_ dv For example, given the homogeneous equation, 2 . ^<^y dy dx dx Substituting y = vx, 2 , a?{xdv + vdx) _ vx^{xdv -\- vdx) dx dx , dv , dx -, whence 1 = av. V X y Integrating and substituting t* = "-, y y log- + loga; = - + c. y Therefore log y — c = -, or Cie^=2/.(logCi = c). PROBLEMS. 1. {x-2y)dx + ydy==0. Ans. log (x - y) - ^—-^ = c. 2. (2y/xy — x) dy -\-ydx=0. Ans. y = ce^i. to m a i.uuci-.4Ki «|"-Q|. _(«) and JdR_ap\pdu_j^du_ (9) \da; 62; y 62 dx Multiplying equations (7), (8) and (9) by 72, P and Q, respectively, and adding, '■(S-f)* 0, and ? > 0. 38- f-T^lo = A logf^^\ ^l^en a > 0, and ^ > 0. J (T — 6 V 2 a6 \a — 6a;y J or — or ^x + a Ja + 6a^ 26 ^V ^/ /' a^dx X fa . fb J x^(a 4- 6ar) aic ^f a^ ^f a 43. f — ^^ = E _j i — arc tanaj\/-. 44 45 ' J (a + 2m-l c?a; + 5a^"*+^ 2 ma (a 4- ^a^)"* 2 ma J (a + bx^ 4- fta^^'+i 2 m6 (a + 6x2)m 2 m6 J (a + bx^) -f- mb J (a dx APPENDIX. 241 Expressions involving (a-^baf). 46 . i OCT (a + bafydx ^m-«+i ^(j _^ bx^'Y^^ — (m — n + 1) a Cx"""" (a + bx^dx ~~ 6 (wp + m + 1) 47 . n»-'" (a + bx^y dx .^-m+i ((J 4. 5a;'*)p+^ + 6 (m — np — n — 1) j a;-"'+"(a + bx'^ydx — a(m — 1) 48. Cx'^ia-^-bxydx a;"'+i (a + 6a;''y + anp | a?"*(a + 6a;'*)^'"^da; np -f- m + 1 49. Cx"^ (a -{-bi '*)-'' dx ar+'\a 4- 60;")-*'+^ — (m + n + 1 — np) j af (a + 6a5")-»^^daj ~" an{p — l) III. IRRATIONAL ALGEBRAIC FUNCTIONS. 50. CVa + bx dx=— V(a + bxf. J ob 51 f-^^= = ?V^T6^. ' J ^a-\-bx b Expressions containing Va + bx, 2 52. J a;VaH- toda; — 2(2a-3&a;)V(a + &a?y ''• ■ -^75^- 36^ ^_2(2a-Mv^T6^. J X V Va + &ic + Va/ 242 55. 56v 57. DIFFERENTIAL AND INTEGRAL CALCULUS. r ^^ =-1- logf^^^^\ when a>0. 6a; 15 6=^ a;^ Va -|- 6a; VaH- 6a; aa; 6 , f^a -\-hx— Va 2 Vo^ VVa H- 6a; + Va, i)- Expressions containing VoM-^- 58. rV^^T^c?a; = |V^M^ + ^'log(a; + VaM^. 59. f— ^=: = log(a;+ V^M^. -^ Va^ + a;2 60. f ^^^ =V^^T^. *^ Va^ + a;^ 61. J^^ dx = V^+^ - a log (« + vj+^). 62. f '^^ =llog — g 63 64. oi^dx X ,.__ = I VSq^ - 1 log (0! + Va^ + 0^). Va^ 4- ar -^ ^ 65 66 •^8 8 / da; _ a; (a2 4- a;2)| a^ V^+^ 67. ^{a"- + af)^ da; = I (2 ar' + 5 a^) V^T^ + ^ log (a; + VoM^. */ 8 8 68 C ^^ ^ VoM^ , 1 w^ ^ + V^H^ ' -'a;«V^M^ 2aV "^ 2 a« ^^^ a; 7 APPENDIX. 243 69. f- dx VoM-^ dx Vci^ + x^ 70 .p a^ + log a; + Va^ + a^. 71. 72. 73. r 74. 75. 76. Expressions containing Va^ — a^. j Va^— a^ c?aj = |- [a;Va^ — a^ + a^ arc sin - ]• /da; . a; -^=r = arc sm — Va^-x" « j ic ^G? — y? dx = — \ ^{p? — ar*)^. f ^^ = i log f ^ ^^. 77. 78. 79. 80. 81. 82. J X X fx" Va- - xFdx = - -V(a2-ar^)3 + ^YaVa^-a^ + a^ arc sin ^\ / dx a^ Va^ — a^ arc sin — 2 a ^/a'-x" V«^-^^^^_Va^-^_arcsin^. f V(a2 - a^)3 da; = J ["a; V(a2 - ar^« + ^ y/oT^ + ?|-' arc sin -l da; _ a? V(a^ — x^Y a- Va^ — ar^ 244 DIFFERENTIAL AND INTEGRAL CALCULUS. 83. I — = — — arcsm-' Expressions containing Var^ — a?. 85. j VaJ^ — (^dx =-\\x Va?^ — a? — o? log {x + Vaj^ — a^)]. 86. r ^^ =log(;a;4-V^^"^=^^. •^ Va?^ — a* xdx 87. r-4^^=vs^^^^. 88 . fa; Var^ - a'dx = i V(a^ - a^)^ 89. fVa^ - a' ^^ ^ Va^ - a' - a arc cos -• J X X 90. r ^^ ^- arc sec ^ a; da; 1 X ^3=:^^ = - arc sec — Va^ - a^ a a 91. f-^^ = ^V^^:=^2^^' log (a; H-Va^-a^); Va^ — a^ ^ c?a; Va^ — a^ go I •^ar^Va^-a' «'^ 93. fa^ Va^ - a^c^a; = |(2 a^ - a')^^^ - a' - -log (aj+Va^-a^. »/ 8 8 /c?a; Va^ — o^^ , 1 „„« o^« ^ 95. jV{x^ - a2)3da; = I ra;V(a;=^-aT - ^ V^^'^^^ + ^ log (a; + V^^^^)]- 96. / APPENDIX. 246 dx X V(ar^ - ay aWx" - a» Jq?(1qc X 98 ■ ^^ 99. ((x" - a^)^ da; = |(2 a^ - 5 a=^) Var^ - a^ 4- ^ log (a; +Var'-a^. «/ 8 8 Expressions containing V2 aa; ± a^. . rV2 aa; - afdx = ^::i^ V2aa;-ar^ + ^ arc ver -. J 2 2 a 01. I — = arc ver ^^ = arc ver -. V2 aa; - ar^ « 0/5. I . = = — V2 aa; — ar^ + a arc ver -• 03 *^ a;^ .^^ _ V2 aa; - ar^ V2 ax — Qi? ax 04. fa; V2^a^^^^da; = - 3a^ + fta;-2g' ^2ax-x' + ^' arc ver ^. J 6 2 a 05 . rV2a.r-ar^ ^^ ^ V2aa-ar^ + a arc ver -. J X a ^g^ rV2 aa; - ^^^^_/2 ax -a^^f, *^ a:^ V Sax' ^ rv,v r c^a; a; — a 07. I j = — =• ^ (2 ax - o^^ aV2 ax - ar» 08. r ^^^ . =- — ^ — *^ (2 ax - x^)^ a V2 ax - x* 09. r ^^ = log (x -4- a-^-V2 ax + x^. ^ V2 ox + x2 246 DIFFERENTIAL AND INTEGRAL CALCULUS 110. I — = ^7- — -^2 aic — ar + I a^ arc ver -. -^V2ax-a^ 2 a lit, r^^^,^ = _/^^+6aa;+5a2^V2^^"^r^ + |a *^ V2 ax — a^ \3 / #^ Expressions containing Va + 6aj ± cxl arc ver ?. 112. r ^^ =— log (2 ca; 4- 6 + 2 V^ VaT6^"+^). »^ Va + 6fl; 4- car^ Vc 4c 6^ — 4 ac log (2 ca; + 6 4- 2 Vc VoT^^T^). = — arc sm c^ 2car-6 VP + 4ac 114. r ^^ =-^ *^ Va 4- 6x — ca;^ Vc 115. I Va 4- 6a; — cMx — ^^ ~ Va 4- 6a; — car^ J 4c , 6^ 4- 4 ac . 2 ca; — 6 4 — arc sm 116. / 8c2 V6'4-4ac a;c?a; Va 4- 6a; 4- car^ Va 4- 6a; 4- ca;^ ^ — ^r J- log (2 ca; 4- 6 4- 2 Vc Va H- 6a; 4- co^) 2cLVc 117. r ^ ^^^ ^V a4-6a;4-c.- ff-|^^ */ Va 4- 6a; 4- car^ V^c 4cV 36^_^\ r c^a; 8 c2 2 c J J Va + 6a; + cy? x'^dx a;"~^ Va 4- 6a; 4- car^ 118 •'^ Va4- 6a; 4- ca;^ ^^ a;'»-2r?a; 2n — 1 hC x'^-Hx _ coi? n-1 a r a;^-^r?a; 2n-\ h C x^'-Hx ^ ' cJ Va 4- 6a; 4- car^ 2n cJ Va 4- 6a; 4- APPENDIX. 247 IV. TRIGONOMETRIC AND TRANSCENDENTAL FUNCTIONS. 119. 120. 121, 122. 123. 124. sin^ xdx = ^x — ^ sin 2 x. j eos^ X dx = ^ X -i- 1 sm2 X. I tan xdx = log sec x. I Gotxdx = log sin «. I ^-^ = log tan 4- X. J smx /; '«* logtanf^ + i^). cosx 125 . j cosec xdx = log tan ^ a;. C dS 2 . f/^a-^M • I ;^ ;; = — arc tan ( -)- J a + 6 cos ^ Va^ — h^ W^ + ^/ ^ tant , when a>b, log V6 -f- a + V^ — a tan - ^^' ~ ^' V^T^-Vft-atan^ when a < 6. 127. 128. 129. 130. 131. j a; sin a; da; = sin a; — a; cos «. j x^ sin a; da; = 2 a; sin a; — (a;^ — 2) cos x. j a; cos X da; = cos a; -I- a; sin x. I ar* cos a; da; = 2a;cosa;H- (ar^ — 2) sin a;. /sin a; -, a^ , x^ oF , 248 DIFFERENTIAL AND INTEGRAL CALCULUS. 33. j arc sin a; da; = ccarc sin a; + VI — ar*. 34*. I arc cos a; c?aj = a; arc cos x — Vl — ar*. 35. j arctana;da; = a;arctana; — •|log(l ■\-y?). 36 . j arc cotan a; c?aj = a; arc cot a; -|- ^ log (1 + a?^. 37. j arcversa;c?a;= (a; — 1) arc vers a; -f- V2 a; — ar'. 38. I log xdx — x log a? — a;. /' dx 11 = log (log a;) + loga; + 2^ log^aj + ^—3-2 log^a; -♦- •• 40. r_i^ = log(logaj). J xXo'gx xe'^^'dx = ~{a' — 1). A o C^ax c\^ «. /i^ «*** (<* sin X — COS a;) 42 . I e"* sin a; aa? = — ^^ — ^« J a^ + l C ay. J e'*'' (a COS a; + sin a;) 43. I e"* COS a5C?a; = — ^^ — ^ ^' 44. Ce^logxdx = C^^^-^ f—dx. J a aJ X 45. fa;«e«*da; = ^^ - - fx'^-V^da;. 46. ra;«loga;c^a; = a;-+fi^---i— -1. 47. rsin"a;da; = - ^^^""^"^^^ + ^^^ fsin^-a^dc.. J n n J m + n m-\- 1 I cos*" X sin"^ xdx. m + n m .n-l APPENDIX. 249 148. I cos*^xdx = -Gos*^~^xsmx-] — ^^— I cos'*~^xdx. J n n J 149. f COS"* X sin" xdx=^ ^^^"'^ ^ ^^^^^"^ ^ + ^^^^^ rcos"'-2a;siii"a;da; »/ m -\-n m-\- nJ 150. ftan" xdx = *?^!ll^ _ Aan-^ x dx. 151. 1 a;"'log**a;c?a; = -^^^log"a; ^ fcc^log'-^ajc^a;. J m + 1 m + 1*/ r^!!^ — a;"*+^ ■ m + 1 r a^^'da; J log" X {n — 1) log""^ ic n — lJ log""^ a? 153. ra"^a;"da; = -^^^^^^^^^ ^ — CoTx^-^dx. J mloga mloga*/ ra'dx a^ logg C a'dx J of (m — Vjof'^ m — lJ a;**"^ -.t.,r r ai « ^ e"*cos"~^a5(acosicH-nsma; 155. I e*''cos"a;daj = ^^ -— ' . or -\-n^ J /rgfn 1 x^ COS ax dx = -— (aa; sin aar + m cos aa;) 250 APPENDIX. Note A. Assume that the surface is given as in Fig. 39, and let the point P in Fig. 45 correspond with the point P in Fig. 39. PX\ PY', and PZ' are drawn parallel to the coordinate axes. Let PS be the section of the surface made by the X'Z' plane, and let PB •^ be the section made P m A a;' by the Y'Z' plane. Let PA' be the inter- section of the plane tangent to the surface at P and the X'Z' plane, and let PJSf' be the line cut from the same tangent plane by the Y'Z' plane. Evidently PA' and PJV' determine the tan- gent plane at P. Now let a plane be passed parallel to the X'Z' plane at a distance d below it. This plane cuts the lines PA' and PN' at A' and N' respectively. A' A and N'N are drawn perpendicular to PX' and PY' respectively, and the points A and M are connected. PC is drawn perpendicular to AN, then CC is drawn perpendicular to A'N', and finally the points P and C are joined. From the equation of the surface by Art. 42, dz_ dx By Let PA = m, and PN — n. and tan^P^' = tan NPN' (1) (2) Now PC :m = n: NA = n ; Vm^ + PC -y/m'^ + n^ Also tan(7PC" = ^= d d Vm^ 4- ^^ mn hence tan2CPC"=— ,H--, PO tan^^P^' + tan^iVPiV^'. But Therefore or CPC' = y. by (1) and (2) g)' STANDARD TEXT-BOOKS PUBLISHED BY D. VAN NOSTRAND COMPANY. ABBOTT, A. V. The Electrical Transmission of Energy. A Manual for the Design ♦-of Electrical Circuits. Fully illustrated. 8vo, cloth $4.50 ATKINSON, PHILIP. The Elements of Electric Lighting, including Electric Genera- tion, Measurement, Storage, and Distribution. Eighth edition. Illustrated. 12mo, «loth $1.50 The Elements of Dynamic Electricity and Magnetism. 'Second edition. 120 illustrations. 12mo, cloth $2.00 Elements of Static Electricity, with full description of the Holtz and Topler Machines, and their mode of operating. 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