GIFT OF Daun:hter of LKGlNE^Rlit AN ELEMENTARY TREATISE INTEGRAL CALCULUS FOUNDED ON THE METHOD OF RATES OR FLUXIONS WILLIAM WOOLSEY JOHNSON PROFESSOR OF MATHEMATICS AT THE UNITED STATES NAVAL ACADEMY ANNAPOLIS MARYLAND NEW YORK: JOHN WILEY AND SONS, 53 East Tenth Street, 1892. COPYPIGMT, 1881, By JOHN WILEY AND SONS. \\\c^ c\ '-^ ^^"\'-' 01 FT OF It. ENGINEERFNG LIBRARY p«ess or i. i. LiTTie & co., M08- 10 TO aa ASTon place, new roRI* S-i PREFACE. This work, as at present issued, is designed as a shorter course in the Integral Calculus, to accompany the abridged edition of the treatise on the Differential Calculus, by Pro- fessor J. Minot Rice and the writer. It is intended hereafter to publish a volume commensurate with the full edition of the work above mentioned, of which the present shall form a part, but which shall contain a fuller treatment of many of the sub- jects here treated, including Definite Integrals, and the Me- chanical Applications of the Calculus, as well as Elliptic Inte- grals, Differential Equations, and the subjects of Probabilities and Averages. The conception of Rates has been employed as the foundation of the definitions, and of the whole subject of the integration of known functions. The connection be- tween integration, as thus defined, and the process of summa- tion, is established in Section VII. Both of these views of an integral — namely, as a quantity generated at a given rate, and as the limit of a sum — have been freely used in expressing geometrical and physical quantities in the integral iorw. 508233 1 V PREFA CE. The treatises of Bertrand, Frenct, Gregory, Todhunter, and Williamson, have been freely consulted. My thanks are due to Professor Rice for very many valuable suggestions in the course of the work, and for performing much the larger share ^f the work of revising the proof-sheets. W. W. J. U. S. Naval Academy, July, 1881. CONTENTS. CHAPTER I. Elementary Methods of Integration. I. PAGE I The differential of a curvilinear area 3 Definite and indefinite integrals 4 Elementary theorems 6 Fundamental integrals 7 Examples I lo Integrals . II. Direct integration 14 Rational fractions 15 Denominators of the second degree 16 Denominators of degrees higher than the second 19 Denominators containing equal roots. . . « 22 Examples II 26 III. Trigonometric integrals 33 Cases in which sin"' cos" d9 is directly integrable 34 The integrals The integrals , and 36 37 VI COiVTEN-TS. PAGE Miscellaneous trigonometric integrals 38 The integration of ; aq " a + 6 cose ^ Examples III 43 CHAPTER II. Methods of Integration — Continued, IV. Integration by change of independent variable 50 Transformation of trigonometric forms 51 Limits of a transformed integral 53 The reciprocal of x employed as the new independent variable 53 A power of x employed as the new independent variable 54 Examples IV 56 Integrals containing radicals 59 Radicals of the fonn /t/{ax'^ -V b) - 61 The integration of ^ — - 64 V(-^ ±0.') Transformation to trigonometric forms 65 Radicals of the form /^[ax^ -ir bx + c) 67 The integrals | — ^ and | ^-^ 68 Examples V 70 VI. Integration by parts 77 A geometrical illustration 78 Applications 78 Formulas of reduction Si Reduction of sin'" 9 ds and cos'" Qd6 82 Reduction of sin'" flcos" e 5- 2 V^. 2 i ^'xdx, f^rl 6. f (.v-i)V:^, ^'_^> + ^_^. •'' 3 f- 3-8 7. r'(« - ^;c) V;c, «V - abx + —1 ':=-'. Jo 3 Jo 3^^ 8. f" {a + ;c)Vx, a^ + ^^ + CA-' + -T = ^a\ J -a 2 4j-c 9- J^ -, 2 log a. to. J — , l0g(-A-)J =l0g2. I.] EXAMPLES. II II. I — —ax, la r 12. £-^dx, J o 13. sin 6dd, ' o 14. COsa:^.x:, J o n4« s. 2 '/.v(a'' + f«:v; + ^x^) I = 23H • a'\ -In ey — I. I — cos sm -V i- dd 16. *■' ^.r 17 dx i»00 /•OO dx V {x" - 1) tan6' Ir'T 19. If a body is projected vertically upward, its velocity after t units of time is expressed by a denoting the initial velocity ; find the space Si described in the time ti and the greatest height to which the body will rise. u =\ vdt = at^ —ig^i\ when V = o ,t= — , s = — - 12 ELEMENTARY METHODS OF INTEGRATION. [Ex. I. 20. If the velocity of a pendulum is expressed by 7Tt V = a cos — , 2T the position corresponding to / = o being taken as origin, find an ex- pression for its position s at the time /, and the extreme positive and negative values of s. 2Ta . 7Tt s = sm — , n 2T j- = ± when t = r, ^T, ^t, etc. 21. Find the area included between the axis of x and a branch of the curve y = sin X. 2. 22. Show that the area between the axis of -v, the parabola y = 4ax, and any ordinate is two thirds of the rectangle whose sides are the ordinate and the corresponding abscissa. 23. Find (a) the area included by the axes, the curve J = ^■', and the ordinate corresponding to a: = i, and (/i) the whole area be- tween the curve and axes on the left of the axis of ^'. 24. Find the area between the parabola of the nih degree, a"-'y = x", and the coordinates of the point {a, a). ft + I § I.] EXAMPLES. 13 25. Show that the area between the axis of x, the rectangular hyperbola xy=Y, the ordinate corresponding to .v = i, and any other ordinate is equivalent to the Napierian logarithm of the abscissa of the latter ordinate. For this reason Napierian logarithms are often called hyperbolic logarithms. 26. Find the whole area between the axes, the curve and the ordinate for x = a, m and n being positive. rr ^ ^ ^^ If n > m, ; if n ^ m, 00 . 27. If the ordinate BJ? of any point B on the circle ;t' +_>;'= a' . • be produced so that BJ? • jRF = a'', prove that the whole area between the locus of B and its asymptotes is double the area of the circle. 28. Find the whole area between the axis of x and the curve y {a' + x') = a\ TtC^. 29. Find the area between the axis of x and one branch of the com- panion to the cycloid, the equations of which are x = atj} y = a {i — cos tp). 27td\ 14 ELEMENTARY METHODS OF INTEGRATION. [Art. II. II. Direct Intcg7'ation. II. In any one of the formulas of Art. lo, we may of course substitute for x and dx any function of x and its differential. For instance, if in formula {b) we put x — ^ in place of x, we have J Z^a = ^og (^ - ^) or log {a - x), according as x is greater or less than a. When a given integral is obviously the result of such a sub- stitution in one of the fundamental integrals, or can be made to take this form by the introduction of a constant factor, it is said to be directly integrablc. Thus, sin mxdx is directly in- tegrable by formula (c) ; for, if in this formula we put mx for 6, we have 1 sin 7nx - in dx^= — cos mx , hence sin in X ■ vidx =^ cos ni x . VI sin mx dx = — J m J So also in \'{a + bx^) x dx , the quantity x dx becomes the differential of the binomial {a + bx"^) when we introduce the constant factor 2b, hence this integral can be converted into the result obtained by putting {a + bx^) in place of ,t'in >/ xdx, which is a case of formula {a), lUS [V{a + bx") X dx = —\{a + bx'f 2bx dx = -^{a ^ bx'Y . Thus §n.] DIRECT INTEGRATION. 15 12. A simple algebraic or trigonometric transformation sometimes suffices to render an expression directly integrable, or to separate it into directly integrable parts. Thus, since — sin X dx is the differential of cos x, we have by formula {U) sin x dx tan X dx ^ cos X = — log cos X . So also, by formula (/), han^ Odd = by (e) and (a), {sec^d- i)dd=tsine -d; |sin=* edd -- by (7) and {a), (I - cos^ d) sin ddd=-cosd + i^ cos^ d ; /(f^V-^ = l7fr^)^^ dx V {i — ^) 2 . (i - ;j^)-* (- 2x dx) = sin-^ x- V {i - x^). Rational F^^actions. 13. When the coefficient of dx in an integral is a fraction whose terms are rational functions of x, the integral may gen- erally be separated into parts directly integrable. If the de- nominator is of the first degree, we proceed as in the following example. r,v^ — ;f + 3 Given the integral by division, 2X -^ \ x^ — X + ^ _x 3 2x + 1 ~ 2 4 dx; + 15 I 42^+1 1 6 ELEMENTARY METHODS OF INTEGRATION, [Art. I3, hence f'^ - ^ + 3 V. _ ^ f w. _ 3 f .. ^ 15 f dx dx = ]\xdx-^[dx +^ [ 2 ,V + I 2 J 4 J 4 J 2X + I X' 7.x IK , , . 44s When the denominator is of higher degree, it is evident that we may, by division, make the integration depend upon that of a fraction in which the degree of the numerator is lower than that of the denominator by at least a unit. We shall consider therefore fractions of this form only. /denominators of the Second Degree. 14. If the denominator is of the second degree, it will (after removing a constant, if necessary) either be the square of an expression of the first degree, or else such a square increased or diminished by a constant. As an example of the first case, let us take The fraction may be decomposed thus : ,r + I A- — I + 2 I + {x-\f- {x-\J - x-\^ {x-\f' hence [ x + \ J { dx [ dx 2 = log (x— l) — 16. The integral f ./' "^ ^ . dx ^ } X- + 2x + 6 §11.] DENOMINATORS OF THE SECOND DEGREE. 1 7 affords an example of the second case, for the denominator may be written in the form x^ + 2x + 6 — {.V +1)^ + 5. Decomposing the fraction as. in the preceding article, ;r + 3 X + I 2 {x+ if+s Gt' + i)'+ 5 ('t-+i)'+5' whence ^ + 3 .^^^ _ f (^ + I) ^'^' ^ o f ^-^' nx+ i)dx ^ J d^ ](x -{- if + 5 Ju' + i)^ x^ -\- 2x + 6 ' J (-1^ + i)^ + 5 ]{x + if + S' The first of the integrals in the second member is directly integrable by formula (b), since the differential of the denom- inator is 2 {x + i)dx, and the second is a case of formula {k'). Therefore ' -^ ■ 3 dx = 4- log x.^^ + 2x + 6) + -— tan"^ — ix" + 2x + 6 ' - ^ ^' ' ' ' ^ ■ 4/5 V5 16. To illustrate the third case, let us take 2x + I j^- dx, in which the denominator is equivalent to {x — \f — 6|-, and can therefore be resolved into real factors of the first degree. We can then decompose the fraction into fractions having these factors for denominators. Thus, in the present example, as- sume 2^+1 A B , ^ "r-—-, .... (I) Pi?' — X —^ X — ^ X + 2 in which A and B are numerical quantities to be determined. Multiplying by {x - 3) {x + 2), 2x + 1 = A{x + 2) + B{x —s) (2) 1 8 ELEMENTARY METHODS OF INTEGRATION. [Art. l6. Since equation (2) is an algebraic identity, we may in it assign any value we choose to x. Putting x = 3, we find y = $A, whence A = ^, putting X = — 2, — 3 = — 5^, whence B = ^. Substituting these values in (i), 2;tr + I _ 7 ^ 3 ,i^-x-6 5(-^-3) SU + 2)' whence l ^^l^le "^"-' = ^ ^°- (-" - 3) + * ^°g ("^ + -)• 17. If the denominator, in a case of the kind last considered, is denoted by {x — a) {x — b), a and b are evidently the roots of the equation formed by putting this denominator equal to zero. The cases considered in Art. 14 and Art. 15 are respectively those in which the roots of this equation are equal, and those in which the roots are imaginary. When the roots are real and unequal, if the numerator does not contain x, the integral can be reduced to the form f dx ]{x-a){x-by and by the method given in the preceding article we find f dx I [x — a) [x — b) a — b log {x -a) - log {x - b) ' log-^" (Ay a-b'^^ x-y * The formulas of this series are collected together at the end of Chapter II., for convenience of reference. See Art. loi. § 11.] DENOMWATORS OF THE SECOND DEGREE. I9 in which, when x < a, log {a — x) should be written in place of log {x — a). [See note on formula {b), Art. 10.] If ^ = — a, this formula becomes { dx I . X ~ a , ... ix^ — a^ 2a X + a Integrals of the special forms given in (A) and (A') may be evaluated by the direct application of these formulas. Thus, given the integral f ^^ l2x^ + ^x ~ 2' if we place the denominator equal to zero, we have the roots « = 1., ^ = — 2; whence by formula (A), r dx _jf ^^ _^^i-^~i. J 2^ + 2,^ — 2~ '^]{x — -|) (x + 2) ""■ 2 ' ij °^ X + 2 ' or, since log (zx — i) differs from log {x — D only by a con- stant, we may write f dx I , 2,r — I = - lof hx^ + ^x — 2 5 ^ X + 2 Denominators of Higher Degree, 18. When the denominator is of a degree higher than the second, we may in like manner suppose it resolved into factors corresponding to the roots of the equation formed by placing it equal to zero. The fraction (of which we suppose the numerator to be lower in degree than the denominator) may now be decom- posed into partial fractions. If the roots are all real and un- equal, we assume these partial fractions as in Art. 16; there being one assumed fraction for each factor. If, however, a pair of imaginary roots occurs, the factor cor- 20 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 8. responding to the pair is of the form (,r — af + /^, and the partial fraction must be assumed in the form Ax ^ B {x-af + fS'^' for we are only entitled to assume that the numerator of each partial fraction is lower in degree than its denominator (other- wise the given fraction which is the sum of the partial fractions would not have this property). 19. For example, given ^ + 3 ^^. )(x'+ l)ix- I) Assume X + 3 Ax -^^ B C , . (.r^ + I) (^ — i) x^ + I X — I whence X + 3 = {x- i) {Ax + B) + {x^+ i) C. Putting x= 1, 4 = 2(7, whence C = 2; putting X = o, • 3 = — B + C, whence B — — i. To determine A, any convenient third value may be given to x; for example, if we put x = — i, we have 2 = ~2{- A + B) + 2C .-. A=-2. Substituting in (i\ ,r + 3 _ 2 2.y + I {x^ -^ l){x —l)'~ X — I ;ir2 + I ' §11]- DENOMINATORS OF HIGHER DEGREE. 21 therefore J(,t-2+i)(;i;-l)^-^~4r- ■ 2x dx dx ,r + I = 2 log {x — i) — log (,1-^ + I) — tan ^ X. 20. If the denominator admits of factors which are func- tions of ,r^, and the numerator is also a function of ,r^, we may with advantage first decompose into fractions having these factors for denominators. Thus, given f x^dx Putting J for x^ in the fraction, we first find y _ + f — a^ 2(j + c^) 2( J — c^) hence x^dx j^ — cc 1 — "S \ Ji dx + dx x^ -\-.a" therefore [see equation {A'), Art. 17], x^dx I , X — a I .X —x 1 = — log 1 tan~ ^ - . XT — a* 4a xj + a 2a a This method may sometimes be employed when the nume- rator is not a function of x^ ; thus, since x' - a' 2^(,r3 - a^) 2a^{x^ + a^ we have x X X^~a^ 2d^ {x^ - a') 2a^ {x^ + a^) ' 22 ELEMENTARY METHODS OF INTEGRA TIOX. [Art. 20. hence xdx I . x^ — (^ = 7:5 ^og Jx* — a* 40^ ^ x^ + a^' 21. The fraction corresponding to a pair of equal roots, that is, to a factor in the denominator of the form {x — a)\ is (see Art. 14) equivalent to a pair of fractions of the form A B + X — a \x — ay we may, therefore, at once assume the partial fractions in this form. We proceed in like manner when a higher power of a linear factor occurs. For example, given f x ^ 2 , ax\ ]{x - \f{x + i) we assume X -^ 2 A B ^ C D + 7 To + : + {x - l)\x + 1) {x- if ^ {x - 1)2 " .1- - I .V + r whence x + 2=[A + B{x- i)^C{x-if]{x + i) + D{x-i)\ . (I) Putting X — I, we have 3 = 2yi .-. A=l. The values of B and C may be determined as follows : if we substitute the value just determined for A, equation (i), is identically satisfied by a- = i, hence it may be divided by x— i. We thus obtain -\=iB ^ C{x- i)-]{x+ i)+D{x-if . . (2) § II.] MULTIPLE ROOTS. 23 in which we may again put ;r = i, whence B = — \, In like manner from (2), we obtain \ = C{x^\)+D{x-\), from which C =^ , and D = —\. Therefore dx •^+^ dx^l. (;ir— l)^(-r+l) 2]{x—\f Af]{x—\) ' dx \ { dx \{ dx _\ (x- 1)^ ~ 4 J (x- lY "^ sji^ 8 X + I 3 I I - ;ir — I 4{x-if^4{x- I) ^8 ''X+ I 22. In this example, after obtaining the values of A and D from equation (i) by putting x = i, and x = — i, two equations from which B and C might be obtained by elimination could have been derived by giving to x any two other values. Con- venient equations for determining B and C may also be obtained by putting ;t: = i in two equations successively derived by differentiation from the identical equation (i). In the first dif- ferentiation we may reject all terms containing (x — i)^; since these terms, and also those derived from them by the second differentiation, will vanish when x = i. Thus, from equation (i), Art. 21, we obtain 1 = A + 2Bx + 2C (x^ —i) + terms containing (x — ly. Putting X = I, and yi = | , we have B = — I. Differentiating again and substituting the value of B, o = — I + 4CX + terms containing {x — i), and, putting ;ir = i in this last equation, C= ^ , 23. When the method of differentiation is applied to a case 24 ELEMENTARY METHODS OF INTEGRATION. [Art. 23. in which more than one multiple root occurs, it is best to pro- ceed with each root separately. Thus given, f :L±J dx ;tr+ I A B C D + +7 T-I^^ + {X - \f {X + 2)2 {X- if' X-l (X + 2)2 ' X + 2 whence x+i=[A+B{x-i)]{x + 2f+[C+D{x + 2)]{x-iy . . (i) Putting X = I, and x = — 2, we derive A = '-, C=-'-. 9 9 Differentiating (i), we have I = 2A (x + 2) + B (x + 2)2 + terms containing {x — i), 2 I whence, putting x =: i, and A =^ -, we have B = . Again, differentiating (i), we have 1 = 2C {x — i) + D (x — if + terms containing {x + 2), whence, putting x = — 2, and C = — , we have D = — . Therefore f -*' + I ^ — _ 2 I J^ , x + 2 ](x-if{x + 2f '''~ 9 {x - I) ^ 9 (a- + 2) 27 °^;»r-l ' 24-. Instead of assuming the partial fractions with undeter- II.] RATIONAL FRACTIONS. 25 mined numerators, it is sometimes possible to proceed more expeditiously as in the following examples : Given dx ; \x^ (I ■ putting the numerator in the form i -^ x^ — x^, we have I ^ _f_l + f!_^ _f__f!__ dx J a-^ J -r ( I + x^\ (i+x^) Treating the last integral in like manner, (dx (dx f X dx 4- locr — !— : 2X? ^ ^ X = - ^ - log ^ + i log (I + ,r2) = Again, given ]x^{\ + ,r) putting the numerator in the form (i + xf — 2x — x?-, we have r I ^ — W^ f 2 + ;r , Hence by equation (^), Art. 17, (^;ir .r I + ^ 26 ELEMENTARY METHODS OF INTEGRATION. [Ex. II, ■I Examples II. dx r_ dx J {a-xY' [ xdx J o 8. (« + w^)' dXy 0. cos' X sin A dXy ■I fcos //9 sin — log (^ - x\ I a — -v \ log (a' + A-). ^3 - (a^ _ ^)l 3 I, a' a - i/{a' - .v^). (a^ + 3-ry 24 (^ 4- //u-y—a" Zm cct 2^ I — COS X ''• „• s,. " > cosec 9 2 12. I sec' 3 A- tan 3-v ^, sec^jjc 1 Jo O §n.i EXAMPLES. 27 17. COS 13, a'"^dx, 14. (f-^— i)Vjf, ^"- {a — x) dx o Vi^ax — x') ' r J o 18. sec*0^e, 19. tan'^rt'^, 20. sec^ .^ tan .r ^ji:, J o 21. 4/ ^JV, It 22. "" cot' 0^0, 4 24. sin (a — 2e) aTe, w log a ' (i + 3sin'jt:)'' 't/(2aA: — a;°) = o. tan 9 + - tan' 9. 3 -tan^'jt: 4- log cos X It ^sec^^l^:..^. 4 Jo 4 a sin~^ - + Via" — x"). a 1 — log 2 i^iaax — x^^ + a vers ^ — cos(«— 29) 2 28 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. 25- 26. 27. 28. 29. 30- 31- 32. ZZ- 34- 35- cos X dx a — b%\xix'' 4 dx ^ tan X ' 6 tan -v ' dx 1 -V log X dx £^ + e- x' dx A-" + I ' X dx Via' - X*) ' dx V(5-3-v')' dx 2 + 5-v- ' dx ... . ^.vy(2A--l)' P + -V + I ' — -7log(^ — ^sin.v). i log 2. log (- log.v) \ log 2. = - log 2. tan- 'f'. 3 tan " '-v' 1 2 sin -5- t ^3 sin V5 I Vio tan" V2 4 Vz^'"'' VZ 1~3^3" §n.] EXAMPLES. 29 36. 37- dx „ 4/(5 -^x- x^y COS ^ f . r i/(^v^ - a') dx L~J-vV 3 2 .r" — «' -/(a — a ) 39- 40.- - .r + 3 [■40: — X - J a-"^ + I dx^ a' (log 2 - f ). 4^; — ^ log (^' + i) — tan " ^ X 40. -^ ^r, AT + log (.r'' — X + I) -\ T-tan '^ ; — 41. [3 ^-dx. 42. 43- 44. .V - 4 (i +^r .r — AT ^, 3 , -r — 2 .a; +-l0g ; 4 °:v + 2 log ^ (2.y + i)' dx 2X + 3 ' r 2Jr + 3 dx^ 45 (2:1: + i) 3^^" + 3 (I - xy ■ X^ — X + 2 log {2X + 3). - log {2X +1) ^ 2 ° ^ 2a- + I • J.r - 3a; 46. + 2 ^;f, a: + log X — 2 X — I X —2ax cos a ■\- a I , jv — a cos a~l'* ^ — '^ '—. tan ■ * -. = -. — . ^ sin a: a szn a 2a sin ot — 'o 30 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. 47 dx 2ax sec a + a dx f 3-v — I 50. . -^ , dx, *^ J -V — A* — 2X ' 51 ^r^il- (-v+2)(Ar+3)*' [ xdx ]x' + X^-2' 53 f jr' — .r + 2 . 55 ^a: ' - ;c' - a: + I ' r ^ 57 2a tan a log X - — r + In. ^(-^' • + 1)1 I , jf — I 4/2 , :r -log—-— +-^— tan-'-7- 6 °jf 4- I 3 V2 2 . j; 4- I I , :r — 2 £. Jf 4- I I 4 °JC — I 2(a: — l)" , {x— i) V{x 4-1) I _, log -^^ — ^^ — 5 tan X. ^ {x' 4- i)' 2 ^^+_£)L+_I_tan->?^^. V3 Vs 6^0g^^_;,+ , 58- J (,_,)^(,^.,,^ > -log(^-i)--log(.+x)-^^^-^^. *l. ^X'-I^J /_,i ~"^/v-v' k3 i^'lnLii^;.i.t->t ''<'r' C'i:^ifi^(£A\UL ^i^* JC 09 §11.] EXAMPLES. 31 f dx \ogx log (i + df) log (i -^ x^) tan - ' x. 1 x^ — x -{■ 1 2 °^a:" + ^ + I 6o- -4—; — 5—; — "^> J :x; + :V + I 61. I ^^f ~.' dx, JlogJc + -log(^- 2) +^log(j; + 3) ] X -V X — tx 6 2 3 62. -^ 1 , ] X — X — \2 -log 1 ^tan-'-^-. 7 ^ a: + 2 7 4/3 ^3- J(p-:r^- 64. 65- 66. ^-"- - 3^ ^^ x' -x" -2' dx {x^ + a^) {x^ by b"" + 1 x—i X 4 ^a: + I 2{x' — i) ^- tan" ' log 2a a 4a jc + a I , a; — 2 6^°s^^TT I r, ^ + 3 , ^ ^ - 1 •^n ^7- L(ZT^ ^)(x^ + by 68 d'dx 2ab {a + b)' 4 — TT 69. —7 — -^rr^, Ja: (i + x') [ dx 4a tan-^.v + log V{i + ^•') 70 I I tan - ' ;v H 1 X s^ 32 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. 71- 72. dx .v(i+.rr' dx log— + X {a + bx') ' I + A- I + A- I , A-' — log 3a a + ^-v' ' I , b . a •\- b^ 3 + — a log i — 74. Find the whole area enclosed by both loops of the curve y-A-'(i-a'). f 75. Find the area enclosed between the asymptote corresponding to -V = a, and the curve ,.2 ,,» ■_ ^a..a ^' ,,' x'f + a'x' — a'f 76. Find the whole area enclosed by the curve ' d'y" = x' {a- - A-'). 77. Find the area enclosed by the catenary 2a . W. r— X j:— 1 the axes and any ordinate. p-r']. 78. Find the whole area between the witch xy^ = 40* {2a — x) and its asymptote. See Ex. 23. ^na. §111.] TRIGONOMETRIC INTEGRALS. 33 III. Trigono7netric Integrals. 25. The transformation, tan'"^ 6* = sec^ ^ — i , suffices to separate all integrals of the form tan«^^^, (I) in which n is an integer, into directly integrable parts. Thus, for example, [tan^ edB = [tan^ B (sec^ ^ - \) dQ tan* B W ^ a jQ tan^ 6 ad. Transforming the last integral in like manner, we have f ^ ^ ,^ tan* B tan^ B [^ . ,^ \\.d.n^ B dB T^i ■ •+ tan<9^6'; J 4 2 J ' hence (see Art. 12) f .- , ,„ tan*^ tan2 6' ' „ " tan^ BdB = log cos B. J 42 When the value of n in (i) is even, the value of the final inte- gral will be B. When n is negative, the integral takes the form [cot« B dB, which may be treated in a similar manner. 34 ELEMENTARY METHODS OF INTEGRATION. [Art. 2^. 26. Integrals of the form [seC^^^ (2) are readily evaluated whcji n is an even number^ thus U&z^dde = f(tan2 + i)2 see ddS = [tan* dsG(?ddd+2 ftan' 6 sed^Sdd + [see* 6 dd tan'' 6 2 tan^ 6 = — - — + + tan 6. 5 3 If n in expression (2) is odd, the method to be explained in Section VI is required. Integrals of the form cosQCOdd are treated in like manner. Cases in which sin'" d cos" d dd is directly integrable. 27. If ;^ is 2. positive odd number, an integral of the form [sin'« d cos" QdQ (3) is directly integrable in terms of sin 6. Thus, sin2 ^ cos' ^ ^^ = sin' 6^ (I -sm^df cos ddB _ sin^ 6 2 sin' 6 s\n^ 9 ~~3 V~^ 7 This method is evidently applicable even when m is frac- tional or negative. Thus, putting 7 for sin d, § III.] TRIGONOMETRIC INTEGRALS. 35 fcos^^ (• (! -f)dy { _^ r . hence f cos^ e __ ^ 2 a_ 2 3 + sin2^ Jsint ^"^^ --^y ' 3-^' ~ ~ 3 ' i/(sin^)- When 7;^ in expression (3) is a positive odd number, the in- tegral is evaluated in a similar manner. 28. An integral of the form (3) is also directly integrable when m + n w a7i even negative integer, in other words, when it can be written in the form rsin'" d do cos"'+^^ 6 tan'" S sec^? B dB, in which q is positive. For example, dB hence -r^^ —^ = (tan Byl sec* B dB Jsina B COS3 ^ J^ ^ = [(tan B)--2 (tan^ ^ + i) sec^ ^^^ ; c dB 2 ^ 2 . 3. n ^~7i — T tans S — - — T-. • jsm-i B cos-^' B 3 tan^ ^ It may be more convenient to express the integral in terms of cot B and cosec B, thus j^^l^ =|cot* B (cot2 ^ + I) cosQc" BdB cot'' B cot^ B 36 ELEMENTARY METHODS OF INTEGRATION. [Art. 28. Integrals of the forms treated in Art. 25 and Art. 26 are in- cluded in the general form (3), Art. 27. Except in the cases already considered, and in the special cases given below, the method of reduction given in Section VI is required in the evaluation of integrals of this form. The Integrals sin2 e dd, and \ cos^ e do. 29. These integrals are readily evaluated by means of the transformations sin^^ = ^(i — cos 26'), and cos^6' = ^(i + cos 2^). Thus [ sin^ edB = I [dd - I [cos 26 dd = ^6 - \s\n2e, or, since sin 26 = 2 sin 6 cos 0, [ sin^ ddd = \{d - sin d cos 6) {B) In like manner [cos2^^^=^(^ + sin(9cos^) [C) Since sin^ d + cos^ d = i, the sum of these integrals is \d6', ac- cordingly we find the sum of their values to be 6. In the applications of the Integral Calculus, these integrals frequently occur with the limits o and ^n ; from {B) and {C) we derive [' sin^d de = [' co^ Odd = \7t. III.] TRIGONOMETRIC INTEGRALS. 37 The Integrals 30. We have do sin e'cos do sin 6 , and r de cos B Csec^edd , • /i /I = -^T TT- = log tan 6'. sin 6/ cos 6^ J tan (9 ^ Again, using the transformation, sin ^ = 2 sin }d cos |^, (^) we have hence f ^^ _r jdd _f J sin ^^ J sin ^6* cos A/9 ~J •secneud tani(9 j^, = logtani^ (^) This integral may also be evaluated thus, J sin (9 ~ J sin Odd sirr sin d dd I — cos^ 6 ' Since sin B dd — — d{cos B), the value of the last integral is, by formula (^'), Art. 17, I , I — cos — log 2 I + cos = log|/l I — COS + COS and, multiplying both terms of the fraction by i — cos B, we have dB , I — cos 7- = log : TT- sm ^ '=' sm B {E) 38 ELEMENTARY METHODS OF INTEGRATION: [Art. 3 1. 31. Since cos d = sin {\7C + 6), we derive from formula {E), By employing a process similar to that used in deriving for- mula {£'), we have also dd . I + sm ^ n = lOR 7{ • cos U ° COS U {F') Miscellaneotis Trigonometric Integrals. 32. A trigonometric integral may sometimes be reduced, by means of the formulas for trigonometric transformation, to one of the forms integrated in the preceding articles. For example, let us take the integral f dd ]as>md -\- b cos 6' a = k cos a, b = k sin a, I r dd Putting we have f rfft T . J ^ sin 6* + ^ cos 6~ k ]sm{d + a)' Hence by formula {E) f dd I 1 . 1 //I \ ; = - log tan -id + a)) or, since equations (i) give k = V(a2 + ^), f dO tan a = —, a (I) ^^ locr \_2^x\ — as\nO + b cos V{a^ + b^) "^ 2 + tan - ' -1 § III.] MISCELLANEOUS TRIGONOMETRIC INTEGRALS. 39 33. The expression sin md sin nd dd may be integrated by means of the formula cos {7n — n) d — cos {m + 7i) 6 = 2 sin md sin n6 ; whence f . ^ . ^ r^ sin (m — Jt) 6 sin (;;? + ;«) (9 , . J 2 {pz — 71) 2 {m + n) In like manner, from cos {m — n) 6 + cos {m + n)d — 2 cos md cos ;«6', we derive f fl ^ ^/3 sin {m -n)d sin (;;^ + 7i) . . \cosm9cosnd ad = 7 ^ 1 — -—, ; — r— • • {2) J 2 (;;z — fi) 2 {m + n) When m = n, the first term of the second member of each of these equations takes an indeterminate form. Evaluating this term, we have { ■ 1 a jQ ^ sin 27i9 f . \sm^7iddd = , (3) J 2 4« \cos^ lid dd = - ^ ' (4) J 2 An o ^ ,^ d sm 2nd and 4« Using the limits o and tt we have, from (i) and (2), whe7i m and n are unequal integers, •IT fir sin 77id sin nd dd = cos md cos nd dd = o; . . (5) Jo Jo but, when 7n and 7i are equal t7itegers, we have from (3) and (4) r sin" 7id dd = r cos2 fiddd =- (6) Jo Jo 34. To integrate 4/(1 + cos ^) dd, we use the formula 2 cos'' ^^ = I + cos 6", 40 ELEMENTARY METHODS OF INTEGRATION. [Art. 34. whence V(i + cos^) = ± 4/2 cos^^, in which the positive sign is to be taken, provided the value of B is between o and n. Supposing this to be the case, we have f V(i + Qosd)de= V2 [cosi^^^ = 24/2 sin \d. For example, we have the definite integral I 4/(1 + cos 6) dB — 24/2 sin - = 2. 4 Integration of -, -r,' * -^ a ^ cos 6* '35. By means of the formulas I = cos'^ ^^ + sinH^ and cos^ = cosU^ — sin'^^^, we have f dd _ f de_ Jrt + ^cos^~ J(^ + /;)cosH6' ^ {a-b) sin^^' Multiplying numerator and denominator by sec^|^, this be- comes sec^^^rt'^ U + ^ + (rt-^)tanH^' and, putting for abbreviation tan \e = yy we have, since \ sed^^ddd = dy, dS f dv [ dS ^ ^ [ dy ]a + bQOsd " ]a ■\- b ■¥ {a — b)f' § III.] MISCELLANEOUS TRIGONO^rETRIC INTEGRALS. 4 1 The form of this integral depends upon the relative values of a and b. Assuming a to be positive, if b, which may be either positive or negative, is numerically less than a, we may put a + b o 7 ='^- a — b The integral may then be written in the form dy a — b ] (^ -]- y^^ the value of which is, by formula [k'), y tan-^-. c {a — b) Hence, substituting their values for y and c, we have, in this case, I -, a~ — T-^ — 79xtan-M A/'^ rtani^ ]a + bcosd ^{a^ —IP') \_y a + b _ . . {G) If, on the other hand, b is numerically greater than a, this expression for the integral involves imaginary quantities; but putting b + a b ~ a the integral becomes ^, • dy the value of which is, by formula {A'\ Art. 17, I , c +y log- c{b~d) ^c-y' 4- FIEMENTARY AfETHODS OF INTEGRA 770X. [Art. 35. L- Therefore, in this case, dd I . V{b + a)+ V{b-a) tdLW^d log dcos6~ ViS^- a^) ^ V{d + a)- V{b-a) tan ^0 36, U c < I, formula (G) of the preceding article gives de 2 . . {G'] f— ^ J I + r cos V {I — i^) tan' j/rT7^^"^^J- • (') Puttiiifi ^i f.tani^=tani<^, (2) and noticing that ^ = o when ft = o, we may write dd (f> COS V {i — c^)' ' ■ . . (3) Now, if in equation (i) we put ^ for 6 and change the sign of c, we obtain It-- d tan 'I + ^ e y j^— ; tan \ \^\ C cos ^ \'\ I — i''^) hence, by equation (2), Equations (3) and (4) are equivalent to dB ^ d(!> I +^ cos (9 |/(i-i^)' ^5) • d(f) dB and § III.] TRIGONOMETRIC INTEGRA IS. 43 the product of which gives (i + ^ cos 6) {i — e cos (j))= I — e^ . . . . (7) By means of these relations any expression of the form f dd J(i + ^ cos^)"' where n is a positive integer, may be reduced to an integrable form. For de dd J ( I + £" cos 8)" J I + £" cos 6^ ( I + ^ cos 6)" - ' ' hence, by equations (5) and (7), 7 — , Zw = ; 2V7; — i (i — ^ cos 6)"-^ d^ . J„ (i + ^ cos6^)« (i/— ^)""Mo By expanding {i — e cos ^)""', the last expression is reduced to a series of integrals involving powers of cos (f> ; these may be evaluated by the methods given in this section and Section VI, and the results expressed in terms of 6 by means of equa- tion (2) or of equation (7). Examples III. 4 , tan' mx tan mx 1 . tan mx ax, — h x. J ^m m 2. IdLXi xdx, A ~ i log 2. f 4 /c , \ ^. tan'(o + «') ,, , 3. I sec* (9 + a) d% 5^ '- + tan (0 + a). 44 ELEMENTARY METHODS OF IXTEGKA TIOX. [Ex. III. 4. sin' fnx dx, 3^ Jo sm 6 sin 3 5 2 . a^ 4 . z , , 2 . ii. - sin^Q sin* en sin « .^ 7 XI 2 35 ■| cos" — 2 COS- 0. 5. sin' cos' G di'O, 6. L'(sin e) cos* S i/ {\ — CC)%x)dx, 2^/2. f ^^ 1 if^ ~l 31. -5 z ,., ■ ., — , -7-tan~M -tan^ . '^ ]a cos';c + b sin .r a/? L« J { dx I , tan ^ J I + cos x ^2 ^2 f dx^ I a -\- b tan ^''' J a' cos" X — b'' sin' :c ' 2 fl!/J ^ a — ^ tan 9 " fsin .r ^jf 1 < , 1 —77 5 ; r-^ — 7, COS"" H cos:rf. V (3 cos jc + 4 sin' jc) ' < 8 ) fsin X cos' :r ^' Via^-b^) '''''" /a — b 2O — Tt tan 4 J _ r' a + b Ua cos x — asm x); the expression is then readily integrated, and A and k so determined as to make the equation last written an identity. The result is 1 f> cos .r + ^ sm a: - ap + bq bp — aq . , , . . , . iix = , , ,.. X + -'. 7.J log (a cos X -\- bsxxi x). 46 . — — -f- , See Ex. \^. ] a -V b XdiXi. X ^ ax b , , , . \ ~i + "IT"; — 7^ *og (a cos a: + ^ sm x). a' + » = ^ sin 54 METHODS OF INTEGRATION. [Art. 42. wc have Transforming again by putting z = y -\- \, the integral be- comes {ds {dz [{z-\f , [ , f , [dz [ ~ J ^ fl'^ = - J ^^rr + 3 J ^^ - 3 J - -f- J = - 7 + 3- - 3 log - - - Therefore, since z =y + i = - + i = X + I X \^{x^-\f~ ' 2X' '^~ X X+I ^^ X A Power of X taken as the New Independent Variable, 43. The transformation of an integral by the assumption, y — x"" (i) is not generally useful, since the substitution X = jj'", whence dx — - y*" ^ dy, n will usually introduce radicals. Exceptional cases, however, * § IV.] THE EMPLOYMENT OF POWERS OF X. 55 occur. For, since logarithmic differentiation of equation (i) gives dx dy , , — = — , (2) X ny it is evident that, if the expression to be integrated is the product dx of — and a function of x^ , the transformed ex-^ression will be dv the product of — and the like function of y. For example, the expression {x^ — i) dx X {x^ + l) ' dv which is the product of -^ and a rational function of ;tr*, becomes X dy, 4yiy+ I) a rational function of y. Hence, decomposing the fraction in the latter expression, we have {x^-i)dx ^i f .r-i ^.^Jio^ (y + 'f x{x^+i) 4}y{y+i)'' 4 ^ y = loi V{-r* + i) X 44. When this method is applied to an integral whose form at the same time suggests the employment of the reciprocal, as in Art. 42, we may at once assume y z^x''^. Thus, given the integral r dx^ ]^X^L2 + ^^y 56 METHODS OF INTEGRATION. [Art. 44. putting y = ;^^~^ whence <^.i- _ dy we obtain ' 1 ° dx _ \r ydy x*{2 -\-x^)~ 3J. 2J+ I y log {2y + l)" - 6"^ 12 _ ° _ 2 - log 3 12 45. The same mode of transforming may be employed to dv simplify the coefificient of — , when this coefficient is not a ^ X rational function of .r". Thus, the integral f dx ixV{x^-^) will take the form of the fundamental integral (/ '), if we put x'=f, whence dx 2 dy Making the substitutions, we have dx _2f dy _ 2 -I y _ 2 -i fx Examples IV. log (2 + .r) + 2 + A- §IV.] EXAMPLES. 57 X dx (i-.r)- 2X — I 2(1 -.v) f x" - .V + I 2X + I log { 2a- + I ) 8(2.1+ i) ° .r^ dx ■X ^v + 2) = log;/ + 4y- 2 l0g2--. dx I + f -^ :tr— (log I + f-'^). 6. )] = ^ log [ ^(■'- '^) + ^(^ - ffl ■ • -W In like manner we have f dx^ f dz __ . _^ z J VV{x-a){(i -x)-\-^ ]^{fi-a-z') - ^ ^'''~ V{P - a) ' by formula (/') ; hence f dx . , ./ X — a ,^. /O METHODS OF INTEGRATION. [Art. 58. It can be shown that the values given in formulas (A^) and {O) differ only by constants from the results derived by em- ploying the process given in Art. 56. Examples V. s/{a — x)-x dx, {a — x)^ (3.V + 2a). : jv{-r +a)-x-'dx, ^ (a + x)^ -^ {a + x)i + —(a + x)l { Xdx 2 3 , , • Jm^' -X^-X+2^X-2\0g{l +YX). [xdx 2 / , ,, . • J^Tf^T^' -^{x-2a)V{x-^a). ■ \ ^x-i ' ^ ^"'" + 2 log (i - Vx). 6. f {a + x)Krdx, ^-^^Y=.-9A. J-a ' ' 7 4 Jo 28 [ dx 2 /2x — a 7- "t;? r\> -tan '4/ ■■ ]x \ \2ax — a ) a f a 8. \\a-x)^^x^dx, _^:l^^_al__2dyy ^16^, Jo 9 7 5 JV'^ 315 J 2x^ — x^ 44 8 § v.] EXAMPLES. yi Jo o 5 Ji lo 40 ''• 1 ^(7-% ^-^^ 2 (1+ :r) V{i - x). Rationalize the denominator. 3«' 2 (jc + a)'2 — 2 {x + iJ)2 t/(^v + tz) + 4/(^ + ^) ' 3 (a - ^) I, V(^' + i) - I ''■[-^v^^^y i^"^ 4 ^ 4/(a;* + i) + i' I i/(-r^ + i) dx V{x' ^1) , i,_ V(a-^+ i)-i '5- I - ' 2 ■*"4'°^v(a-^ + i) + r 16. (a'^+i) (.r'^- i)t ^x, nV- $ 3 J f'^ ^V.r / , a'"]''^" rS 11^2-1 , 4/(a:'' — a") — « sec - ' - . 72 METHODS OF INTEGRATION. [Ex. V. f x^dx ■ ]>/{x'^ay — . 3 ^dx. See formulas (L) and {K). v' a- xt" _ /T* . "1 -xVix' + a') - - ^Mog [.V + ^{x' + a')] . 2 2 5. I -^^—^. dx, a log ^- + Via' - x ) f dx '■ ]x + vix' ^-dy ^ Vix^ + a^) + '-log [.r + Vix^ + a')] - ^ log [ Vix' + a') + x]- ^^^ '- f dx ^- ]Vix' + 0") -a ' log [ Vix' + fl') + ^J - ^ ^ - or x 24- f-77^T^- -S"^^ Formula {K). -log [a:' + V(i + ^-*)]. §v.j EXAMTLES. 73 25. /^{ax^ + b) dx, \a >o] Fui Viax" + b) = z —x Va. — -log LxVa + Viax^ + ^)] + - xViax' +b). 2 ya 2 26. J {a + a-) \/{x' + b-')' I j^ ^ + V{x' +b') + a - Via' + b') 27 V[a'+b') ^.r + Vi^r' + />') + a + V{a' + b') dx V(i + x^) — cot^ = V3. 29. x^ dx (-v=-«*)t' 30- J {x), are known, and also the successive derivatives oif{x), we shall have \f{x) ^ (,r) dx = f{x) , (x) - |/' (.r) ^^v) dx = /(.r) ^^ (.r) -y ■'(''^) ^^ (x) + j/" (.r) ^,^ (x) dx. Continuing this process, and writing for shortness f, n • • • foi" f{x), (f)^ [x) . . . we have /{x){x)dx=/.^-~/'.<^^^ + + (- \Y"f"~'(t)n + {-lY f"4n'dx. The application of this formula is equivalent to the use of a formula of reduction. Thus the value of x"^ £"^ given in Art. 65, may be derived immediately from it. go METHODS OF INTEGRATION. [Art. 75. Taylor s Theorem. 75. If, in the formula of the preceding article, we put f{x) =F' {Xo +/i- x), and (x) = i, Xo and /i being constants, /' (x) = - F" {xo+ /i - x), f" {x) = F"' {x, ^ h- x\ etc. ; x^ x^ and , (x) = X, ^„ (.r) = — , ^,^ (x) = ^-^ , etc. Hence \F'{Xo+/i — x)dx=.F' {xo + /i — x)'X-\- F" {xo + h — x) -— + F"^' (Xo +/i- x) — :^ dx. Now \F' {xo + h — x) dx = — F {Xo + h — x) ; hence, applying the limits o and /i, we have Fix^ + h)^ F{x,) + /' ' {Xo) h + F" {x:) 1-^ + Jo I-2- • ■ n This formula is Taylor's Theorem, with the remainder expressed in the form of a definite integral. VI.] EXAMPLES. 91 J o Examples VI. a: sin"^ jc + v'(i — .v'^) 7t 2 2. sec~'jc ^/O = - . O ■v' tan - ' A - .v' log(i + .v^) 3 6 "^ 6 16. ■7T 2 I . 2 + A -v' sin- ' A- dx. - A-' sin - ' a- + ^ a/(i — a') 3 9 J„ 6 9 17. e-'' cosxdx. f "-^ (sin A- — cos.x)' 18. U •" '^'" ^ cos a: rt'AT, COS /3s^^^"P sin {ft + x). 19. f - ^ sin' .V ^/a- = - f--^ (i — cos 2a:)^a: , J L 2 J J (cos 2Ar — 2 sin 2a: — 5). I: 20. f* sin a'S, — (sin — cos 0) 2 4 _ i_ 2 §VI.] EXAMPLES. 93 f^ sin X cos X dx, 22. I sin^ ;«9 fl?i9, — (sin 2x — 2 cos 2X). lO ^ ' sin^ m() cos ;;zQ 36 3 sin we cos »«0 4;// Zm 23. Derive a formula of reduction for Xi^ogxY x"' dx^ and deduce from it the value of (log x)" X'" dx = (log x)" ~ (log x)"-^ X'" dx. J(log.^•)^^-Va• = (logaf -- - (logA)^-- + -^ 24. .V cos^ A- dx, 27 4 a" sin x cos .r — i sin' x + ^^. ■'■I 25. jv- sec" ' a'^/jc, .r' sec - ^ A- .r ^/(x'' — i ) log [x + V(.r' — i )] >6. Derive a formula of reduction for Lv« sin {x + a) dx, and de- duce from it the value of X cos X. X" sin {x + a) dx = — x'^sin \ x + a -] — + n a«"^ sir Lv' cos X dx = {x'' — 20^' + 120^) sin a: + {^x* — eox"^ + 120) cos x. 7t X -\- a ^r - 2 94 METHODS OF INTEGRATIO.W [Ex. VI. f - . , , sin' COS sin OcosO , i r, . ^ ,-, 27. cos' sin* 7r 96 J sin' e ' J cos ' fsin' G ^ J cos* flcos'o ^ 34- -^ ^'^^ 31 32 33 cos' ^J _Z cos 3 log tan 4^ 2 sin'' 2 2 sin e 4 cos e 8 cos sin I , — -log tan L4 2 J sm 5 sm t; r • t i ^ — + ^ [G — sin cos 0] 3 cos 3 cos 2 cos sin TT IT 2 f2 - 5 cos IT J >r O^G 48 — IS^T 32 35 36. j ^G + cos 0)' ^0 sin G cos G 4 d^)' _ 2 cos* G' 3 * 5- H + log tan - . 3 cos cos G 2 §VI.] EXAMPLES. 95 37 J sii M sm 6 sin 20 4 sm Q cos 38. Prove that when n is odd 3 cos 63, ^ ■ o + 5 logtan- 8 sin e 8 ^ 2 J six ^9 _sec«-^0 sec«-5G I sin 6 cos" n — X // — 3 and when ;2 is even da sec"-^0 sec"-^© -I 1_ I sin cos'' n — 1 n — 2, + log tan S ; + log tan ■ Z9- \- ■ J si do 40. 41- ^ -+5 sin"'&cos*0' 2 sin" 6 cos'^O 2 'sec h sec + log tan L 3 2 ] V'l^v^- i) , Put X = sec U-y(a-^ - i) + ilog [.r + Vix' - i)]. {a' — x')' dx, ^ '—-^ ^ + iL_ sm - 1 - cos* ^0 = ^ r- . 44- 45- J(-^-^ + i)" V{x' - a') + log [x + ^{x' - a')]. x{x^ — 1) , tan"'ar 8(x' + iy^~8 ■ 96 METHODS OF INTEGRATION. [Ex. VI. , fcos'G — sin'O j,V[i,-, ,\ cosO — sinO 46. L . vJ '^'^ = (i + sin cos J/);^ ^ J(sinG + cos 6)' L J ,^0], (sinO + cosO) sin cos 5 sine + cosO 47. Derive a formula for the reduction of \x%Q.z^xdx\ and refer- ring to Ex. II, thence show that this is an integrable form when n is an even integer. Give the result when // = 4. -v sec" -v dx = a- sec" - ^ -v tan x sec" - ^ :>; « — I X sec* X dx = {n-i){n- 2) n — I J X sec* X tan x sec' x 2 V sec"-''x dx. ^-. — h - [-v tan .V + log cos .rl. 6 3 48. Derive a formula of reduction for from it the value of v cos^ x dx. X cos" X dx, and deduce , .vcos""' .V sm .V cos"-v «— i f .V cos" -v dx = H r. 1 .r cos" ~ ^ .V dx. n n n ) 3 , .vsin.r. , . cos.r. -v cos -v dx = (cos X + 2) -\ (cos- X + 6). 3 9 49. Find the area between the curve _y = sec " ' .r, § VI.] EXAMPLES. 97 the axis of .v, and the ordinate corresponding to .v = 2. 27C — - log [2 + rj] = 0.77744. 50. Find the area between the axis of .v, the curve y = tan" ' -v, 77" l0£ 2 and the ordinate corresponding to x — \. e— = 0.43882. VII. Definite Integi'als. 76. Before proceeding to transformations of definite inte- grals involving the values of the limits, it is necessary to resume the consideration of the relations between a definite integral and its limits, as defined in the first section. By definition, the symbol • a denotes the quantity generated at the rate while X passes from the initial value a to the final value X. The rate of x is arbitrary, and may be assumed constant ; but in that case its sign must be the same as that of the mcrement 98 METHODS OF INTEGRATIOX. [Art. 76. received by .r ; that is, the sign of dx is the same as that of X- a. These considerations often serve to determine the sign of an integral. Thus f" sin xdx . , , . . . , sm X denotes a positive quantity, because dx is positive, and is positive for all value;; of x between o and n. 77. Now let F {x) denote a value of the indefinite integral, so that d{F{x)\=fKx)dx', thus/(,r) is the derivative of F{x). Then, supposing F (x) to vary continuously as x passes from a /^ X ; that is, to have no infinite or imaginary values for values of x between a and X^ the integral is the actual increment received by F {x), while x passes from a to X. In this case, therefore \^'f{x)dx = F{X)-F{a) (I). If, on the other hand, there is any value, n-, between a and X, such that F{a)=.^, equation (i) does not hold true. For example, [dx _ [dx I X and in the case of the definite integral dx 1? § VII.] DEFINITE INTEGRALS. 99 X passes through the value zero, for which F (x) is infinite ; %ve c CHI not therefore tvrite dx __ I Z2 ~ X ~ — 2. This result indeed is obviously false, since dx is here positive, and x^ is never negative for real values of x. The value of the integral is in fact infinite, since the increments received by , while x passes from — i to o, and while x passes from o to I, are both infinite and positive. 78. Since the derivative of a function becomes infinite when the function becomes infinite [Diff. Calc, Art. 104; Abridged Ed., Art. 89], we can have F {a) = 00 only when /{a) = 00; but it is to be noticed that Fix) does not necessarily become infinite wh.en/{x) becomes infinite. Thus, in f' dx /{x) —x-^i, which becomes infinite for ;r = O, a value of x between the limits ; but since Lr-7 dx = |;r^ the indefinite integral F{x) does not become infinite. There- fore equation (i) holds true, and dx 3 „-l3 9 . ,a4 2 J_, 2 79. We have, in the preceding articles, assumed that the independent variable varies uniformly in passing from the lower to the upper limit ; but when a change of independent variable is made, the new variable does not generally vary 100 METHODS OF IXTEGRATIOX. [Art. 79. uniformly between its limits. It is, however, obvious, that, in equation (i), Art. ']'J^ x may vary in any manner whatever in passing from a to A', provided tliat F{.r) remains tliroughotit a continuous onc-valucd function ; x may even pass through infinity, provided F {x) is finite and one-valued when .«■ = 00 . Mtdtiplc- Valued Integrals. 80. When the indefinite integral is a multiple-valued func- tion, a particular value of this function must of course be employed, and it is necessary to take care that this value varies continuously while x passes from the lower to the upper limit. In the fundamental formula (_;') it is sufficient (provided the radical t(i — x^) does not change sign), to limit the meaning of the symbols sin~'-r and cos" 'a- to the primary values of these symbols (see Diff. Calc, Arts. 54 and 55), since these values are so taken as to vary continuously while x passes through all its possible values from — i to f i. 81. In the case of formula {k) the primary value of tan"' x is so defined that, as x passes from -co to + 00 , the primary value varies continuously from — |^rr to + l^r. We may thcre- fo're employ the primary value at both limits, unless x passes through infinity^ as in the following example. Given the inte- gral f? de [i ^cc^Odfi < ]^cos^d -\-gs\n^d J^i + Qtan^d*' if we put tan 6 = .r, this becomes " 3 dx I ^ 5- = - tan -'3.1- i + <^ 3 = - [tan-'(- « 3) —tan-' o]. J But here it is to be noticed, that, as 6 passes from o to ^rt, x § VII.] MULTIPLE- VALUED JJ^^ECMLt^ ] ] -/ ] \ >' \ /\'^^^ passes through infinity when 6 = |7r. Hence, if the value of tan~'3,i' is taken as O at the lower limit, it is to be regarded as increasing and passing through Itt, when ^'=00, so that its value at the upper limit is f tt, and not — ^tt. Hence f6 d6 2n Jo cos^ 6^ + 9 sin^ 6 g ' 82. When the symbol cot-^ x is employed, the primary value, defined in the same manner as in the case of tan~'x, cannot be taken at both limits when x passes through zero. Thus, using the second form of (-^), Art. 10, we have !.: = cot-' I — cot-'(— l), + ^ in which, if cot"' i is taken as -] ;r, cot-'(— i) must be taken as I 7t. Thus dx I -„ = — - ;r. I I ^ x" Formulas of Reduction for Definite Integrals. 83. The limits of a definite integral are very often such as to simplify materially the formula of reduction appropriate to it. For example, to reduce x^ e -"^dx. we have by the method of parts {x"s-^dx = — x"* £-^ + n U--'' x"-'dx. 102 ', I i/f ',',: '■' ''yMETHOJlS. QF INTEGRATIOK. [Art. 83. Now, supposing n positive, the quantity ;«•'' f""^ vanishes when ;jr = o, and also when a' = 00 [See Diff. Calc, Art. 107 ; Abridged Ed., Art. 91]. Hence, applying the limits o and 00, Jo Jo By successive application of this formula we have, when n is an integer, x" £-■'' dx = n {h — i) 2 • I . J o 84. From equation (i) Art. 66, supposing ;n > i, we have f' sin'« edd = ^.^^-^^ \" sin"'-- OdS. Jo ^« Jo If m is an integer, we shall, by successive application of this fl 7t formula, finally arrive at dO = - or Jo 2 as 7n is even or odd. Hence sin 6 dd = I, according if 7n IS even, sm'" dd = ^ ^-^ ^-^ , . . (P) J^ m{m ~ 2) 22' ^ ' and if m is odd, p sin'" dd = ^''' ~ /X 'l^Il^hlUL^ , • - {P') Jo m{m — 2) I § VII.] FORMULAS OF REDUCTION. IO3 •85. From equations (3) and (4) Art. 69, we derive sin"'(9cos"-^/9^«9, sin'« e cos" BdB= ^ m + ;/ . and I ' sin'« B cos" B dB = ^^ ~ ^ V sin'«-^ B cos" B dB. Jo ^^l- + ^^ Jo By successive application of these formulas, we shall have for the final integral one of the four forms dB, s'm B dB, rcos^<^6', or | sin^cos^^^. Jo Jo The numerator of the final fraction ( or ) is in each \m + n ni + 71 J case either 2 or i. In the first case, the value of the final inte- gral is I TT, and the final denominator is 2 : in the second and third cases, the value of the final integral is i, and the final denominator is 3 : in the fourth case, the value of the final integral is \, and the final denominator is 4. Therefore (since the factors in the denominator proceed by intervals of 2), it is readily seen that we may write Fsin" e COS'. Ode = («-)('«- 3)- ■•■('' -0(>'- 3) ■■• ,, . (g) j^ {m + n){m + n — 2) provided that each series of factors is carried to 2 or i, and a is taken equal to tmity, except when m and n are both even, in which case a = ^TT. 104 METHODS OF INTEGRATION. [Art. 86. Elementary Theorems Relating to Definite Integrals. 86. The following propositions are obvious consequences of equation (ij, Art. 77. ^f{x)dx^-^fi,x)dx (I) \ f{x)dx=\ f{x)dx +\ f{x)dx Ja * a i c Again, if we put x — a + b — s,\\q have fV W dx=- \^f{a + b-z)dz=( f{a + b- z) ia Jb ia . . (2) dz by (i), or since it is indifferent whether we write 5- or ;r for the variable in a definite integral, ^f{x)dx= (f{a-Vb-x)dx .... (3) If « = c, we have the particular case ^f{x)dx=tf{b-x')dx .... (4) § VII.] DEFINITE INTEGRALS. IO5 87. As an application of formula (4), we have JT 77 IT J' cos"' Bde^^ cos"Y| - B\ dO = y sin"' Odd . . . . (i) value of Hence the value of cos'" dd as well as that of sin'«6^^/^ is given by formulas (P) and (P). The values of these integrals are readily found when the limits are any multiples of A rr. For, by equation (2) of the preceding article, we may sum the 7t values in the several quadrants. But, putting 6 — k — h 6', and employing equation (i), we have 1^ 'sin«^^^=±| ' cos-^^^= ±]%in'«6'^6', ... (2) /.— in which the sign to be used is determined by that of sin" or cos"' 6 in the given quadrant. In like manner the value of the integral in formula (Q) is numerically the same in every quadrant, and its sign is the same as that of sin"' 6 cos" 6 in the given quadrant. Change of hidependent Variable in a Definite hitegraL 88. It is often useful to make such a change of independ- ent variable as will leave unchanged, or simply interchange, the values of the limits. As an illustration, let us take the definite integral ■" log.r , IL = 2 dx. )ol + X -^ X^ io6 METHODS OF INTEGRATIOiW [Art. 88 If we put X — - , whence log x = — log j, and cix = — 7 r logy u = - , — \!' Bdx (2) The values of the integrals in the second members of equations (i) and (2) are A {b — a) and B {b — a) respectively. There- fore, if we assume ^f{x)dx = M{b-a), (3) we shall have A > M> B. The quantity M in equation (3) is called the mean value of the function /"(-t-) for the interval between a and b. 98. Let b — a =1 n rix\ (4) then the w + i values 6f x, a, rt + A ;r, rt: + 2 A ;f , • • • • b, define n equal intervals into which the whole interval b — a \s separated. Let Xi, Xn, x„he n values of x, one com- prised in each of these intervals; also let 2^/{Xr) Ax denote the sum of the n terms formed by giving to r the n values 1 • 2 • • • • 7/ in the typical term/(-tv) Ax; that is, let 2i f{Xr) AX = f{x,) A 4.- + /(xg) AX"'- + f{x„) A X. . . (5) § VII.] AN INTEGRAL THE LIMIT OF A SUM. 1 13 We shall now show that when n is indefinitely increased the limiting value of 2if{xr) Aa' is f{x) dx. 99. If we separate the integral into parts corresponding to the terms above mentioned ; thus, eh C'i + A-r fa + 2i^x f{x)dx^ f{x)dx+ f{x)dX'-.' J a ■^ '' Ja -i- L .1- + f{x) dx, b - A. and let l/j, M^, • • • • Mn denote the mean values of /(,r) in the several intervals, we have, in accordance with equation (3), Art. 97, [ fix) dx ~ Mi^x ■^M^t.x + Mn i\x (6) Now, since f{x,) and M,. are both intermediate in value between the greatest and the least values oi f {x) in the inter- val to which they belong, their difference is less than the dif- ference between these values oi f{x). Therefore, if we put f{Xr)^AIr + er, (/) er is a quantity whose limit is zero when n, the number of intervals, is indefinitely increased, and i\x in consequence diminished indefinitely. Comparing the terms in equations (5) and (6) we have, by means of equation (7), :S*/(a') ax = [ /(-^') dx + {e^-\- e^ + e„) Ax. ... (8) 114 METHODS OF INTEGRATION. [Art. 99. Denote by e the arithmetical mean of the « quantities rj, ^2. • • • • ^n ; that is, let «r = ^1 + rj + rg r„ ; (9) then, ^ince e is an intermediate value between the greatest and the least value of Cr, it is also a quantity whose limit is zero when 11 is indefinitely increased. By equations (9) and (4), equation (8) becomes ^\f{x,) A.t- = f f{x) dx + e{b- a), whence it follows that /(.r) dx is the limit of -^„/ (x,) dx when n is indefinitely increased, since the limit of c is zero. 100. It was shown in the Differential Calculus, Art. 390 [Abridged Ed., Art. 193], that, in an expression for the ratio of finite differences, we may pass to the limit which the ex- pression approaches, when the differences are diminished with- out limit, by substituting the symbol d for the symbol a. The theorem proved in the preceding articles shows that, in like manner, in the summation of an expression involving finite differences, we may pass to the limit approached when the differences are indefinitely diminished, by changing the symbols -2" and A into and d. The term integral, and the use of the long s, the initial of the word sum, as the sign of integration, have their origin in this connection between the processes of integration and sum- mation. VII.] ADDITIONAL FORMULAS OF INTEGRATION. II5 Additional FoJ'imdas of Integration. 10!. The formulas recapitulated below are useful in evalu- ating other integrals. {A) and {A') are demonstrated in Art. 17; {B) and {C) in Art. 29; {D) and {E) in Art. 30; {F) in Art. 31 ; (Q and {G') in Art. 35 ; {H) and (/) in Art. 50 ; {J) in Art. 51 ; (A') in Art. 52 ; (Z) in Art. 53 ; (J/) in Art. 55 ; (A") and {O) in Art. 58; {P) and {P') in Art. 84; and (0 in Art. 85. dx loc (.1- - a) {x - b) a -b "= X ~b dx I , X — a lot 2a 'X -^ a s\v? 6 do - l{8 - sin 6 cos 0) . . . . . 9 = |(^ + sin ^ cos ^) . . o o . COS' o'^ sin 6 cos ^ losf tan -, n 1 I ~ cos . ^ = log tan 4c/ = log r — -— sm ^ ^ ^ ^ sm ^ ^6* cos dd = log tan TT ^~ + — U 2 _| = losf I + sin ^ cos 6 + 6 cos 6 V{a^ - ^) tan" / 7 tan 4i a + b II J METHODS OF INTEGRATIOX. L^l't. lOI. dd , Vi^ + a) 4- V{d — a) tan J ^ ,^,. \a+dcose Vip'-a^) ^ ^/[b^a) - V{d - a) tan ^ 6 dx 1 ,1- v^ {x^ + rt'-^) rt: — - \oz -^ (^j dx 1 , a — \'(c^ — x^) - lor X \/{c^ -^) a "^ X (/) dx ^{ax'-^bf- bV{a^^+ b) {y) 1 dx Vi-r" ± a") log [_x + i/(.t-2 ± ^)] (>^) 1 1/(;.^ ± a') dx = ^^(-^''^^^') ± l' log [x + VU"" ± ^=)] • • (L) I .r ;f Vf^z^ — .v^ s/ id} — x^) dx — — sin - ' - + ^ ' 2 a 2 (J/) dx ^\{x-a){x-p)\ = 2 log [ V(-r - ^0 + i/f.r - ^^] . . . {N) dx ^l{x- a)^li - x)\ ^''^"^ li (0 sin'" e do = m{m — 2) 2 2 § VII.] ADDITIONAL FORMULAS OF INTEGRATION. WJ sin"' 6 dB = cos^'Odd^"^ ^^ ^^' . . . {p'\ in {in — 2) I ^ ' sin' 6 cos" Odd — ^ ^^ ,, -^^ ^^ — r — ^-^ ^ a, . (0) {in + n){in + n — 2) ^""^ in which a = i, unless m and n are both even, when a = Examples VII. a + d cosQ' [a > d, and n an integer] nir V{d' -b') 2. •2«7r±- ^g 2 + COSS' 2n7C±\7t '■1 3. sm Ode, 55 32" 4- sin^ do, 16 15 „ cos" QdQ, 16 35' !>' 6. sin^ cos^ Q (/9, 37r ii8 METHODS OF INTEGRATION. [Ex. VII. 7. sin' cos^ 6 ih. It 8. I sin"' cos'" ^/G, f X-" dx x^i^a' — x^y dx, 1 -P 2'"Jc 3_ 35 sin"' r/O , 1-3-5 • • (2« — l) TT 2-4-6 • • ■ 2n 2 2-46 • • 2/1 3-5-7- • • (2;/ +1)* 2a' ^3 i6a ' 13- 14 .r^ dx .v' dx 15. Prove that TT 12a .v''-'(a — -r)"'-' dx — I .v'«-' {a — x)"--" dx , Ju o and derive a formula of reduction for this integral, supposing « > and m > i. It ffi — if'* X" -^ {a — x) '" - ' dx = A" {a — x) '" -^ dx. § VII.l EXAMPLES. 119 16. Deduce from the result of Ex. 15 the value of the integral when 7n is an integer. X" - 1 {a — x) '« -■" dx = •J \ ' ^m + n-i 17- n{n + i) • • •{n + m — i) a (a + xy (a — x-y dx. See Ex. 16. a 8. sin' (cos ^y do. Put sin^ Q = x, a?td see Ex. 16. a * , 45045 5-7-II-I9 19. Show by a change of independent variable that x'^ dx dx and therefore \x log A- dx o {a' + xy -Jo {a' + xy ' f"' X' dx _ £ r dx __ 7t_ Jo (a" + X'y ~ 2]od' + .V'~ 4a' ■ Jo {x'+ay ' r tan-^v. dTjc ■ Jo^* + ^ + i' '•i; tan — x dx 4 I _4 > 4 I • a X + a logo. 2a" 6V3- i6d'' 23. Derive a series of integrals by successive differentiation of the definite integral | ?-"•'' dx. X" €-'^ dx = I-2- • • n 120 METHODS OF INTEGRATION. [Ex. VII. 24. Derive from the result of Art. 63 the definite integrals £- ""^ %\Xi nx dx = —f-^ — », and i- "'"^ 0.0% nx dx = —^ — ^\ Jo m + n ' Jo /// + n and thence deri e by differentiation the integrals ' |"00 -3i 3 3 f „ ,„, . , 2w,v ,1 , m — n xe- "'' sin fix dx — -~r. ij-. , and xe - '«■* cos nx dx = t^, rr» ' Jo (/«■ -;- 11) Jo (;«' + r ) 25. From the results of Ex. 24 derive X' €- """ sin ?2X dx = -~^. J— : |"W / 3 _ ..3 x'' e- "'-^ cos nx dx 26. From the fundamental formula {H^) derive and thence derive a series of formulas by differentiation with refer- ence to oc. r' dx _ ^ i-3'"*(2« — 3) I Jo(« +y».vy ~ ^ri-2 •••(«- !)■««-* 27. Derive a series of integrals by differentiating with reference to yS, the integral used in Ex. 26. f" X^*^-^ dv _ 7T I-3-5 • • (2« — 3) I Jo (a + fix^Y ~ ^^ 1.2-3 • • • («- i) * /:;''"* ' § VII.] EXAMPLES. 121 28. From the integral employed in examples 26 and 27, derive x" dx the value of , , ,^ ,,, Differentiate tiaice with reference to /3, and once ivith reference to a. p X* dx _ i'3-i 7t Jo {a + l±xy ~ TTa^ ■ ^eaifi^ ' 29. Derive an integral by differentiation, from the result of Ex. II., 67. dx _ 7T {2a + b) {x' + H') {x' + a')' ~ \a'b (a + bY ' dx 30. Derive an integral by integrating 7t o «" + x^ 2a tan- ' ^ — tan- ' -^ — = - log =^ . .V a.-J ,v 2 g 31. Derive a definite integral by integrating e - '"X sin ;?ji; rti'j; with reference to n. ■ (cos ax — cos /7jc) dx = — log 2 m + a 32. Derive a definite integral from the integral employed in Ex. 31 by integration with reference to ;;/. s-"-^ — s-^"^ \ dx ~ Jo ^- L J L tan ~ ' - — tan n ll- 122 METHODS OF INTEGRATION. [Ex. VII. 33. Derive an integral by integrating with respect to »i m £-ti!x (,Q5 fix dx =: — r- e-n.r _ e-ix I b' + fi" COS nx dx = — log -5 ; 1 34. Derive an integral by integrating with respect to n the integral used in the preceding example. rE-"'^ , . . , N , Ma - b) (sm ax — sin bx) dx = tan" ' — , r . Jo A' m + ab 35. Show by means of the result of Ex. 32 that 1: sin nx , TT dx = — . X 2 36. Derive an integral by integration from the result of Ex. II., 67, 1> tan -•^-tan-'-^ I r: .. = -^ log 37. Evaluate log -^ w^^ by the method of Art. 96. Jo .V + ^ 7r{a- b). 38, Evaluate log •' I + — 3 \ogxdx. 7ta (log a — i). § VilL] PLANE AREAS. 1 23 CHAPTER III. Geometrical Applications. VIII. Plane Areas. 102. The first step in making an application of the Inte- gral Calculus is to express the required magnitude in the form of an integral. In the geometrical applications, the magni- tude is regarded as generated while some selected independ- ent variable undergoes a given change of value. The inde- pendent variable is usually a straight line or an angle, varying between known limits ; the required magnitude is either a line regarded as generated by the motion of a point, an area generated by the motion of a line, or a solid generated by the motion of an area. A plane area may be generated by the motion of a straight line, generally of variable length, the method selected depending upon the mode in which the boundaries of the area are defined. An Area Generated by a Variable Line having a Fixed Direction. 103. The differential of the area generated by the ordinate of a curve, whose equation is given in rectangular coordinates, has been derived in Art. 3. The same method may be em- ployed in the case of any area generated by a straight line whose direction is invariable. 124 GEOMETRICAL APPLICA TIONS. [Art. 103. Fig. Let AB be the gencratiiii^ line, and let R be its intersection with a fixed line CD, to which it is always perpendicular. Suppose R to move uni- formly along CD, and let RS be the space described by R in the interval of time, dt. Then the value of the differential of the area, at the instant when the generating line passes the position AB, is the area which would be generated in the time dt, if the rate of the area were constant. This rate would evidently become constant if the generating line were made constant in length ; and therefore the differential is the rectangle, represented in the figure, whose base and altitude are AB and RS ; that is. it is tJic product of the generating line, and the differential of its motion in a direction perpendicular to its length. 104. In the algebraic expression of this principle, the inde- pendent variable is the distance of R from some fixed origin upon CD, and the length of AB is to be expressed in terms of this independent variable. When the curve or curves defining the length oi AB are given in rectangular coordinates, CD is generally one of the axes; thus, if the generating line is the ordinate of a curve, the differential \s y dx, as shown in Art. 3. It is often, how- ever, convenient to regard the area as generated by some other line. ■ For example, given the curve known as the witch, whose equation is y X — 2a j^ + 4/7lv = (i) This curve passes through the origin, is symmetrical to the axis of X, and has the line .v = 2a for an asymptote, since X = 2a makes y = ± 00 . Let the area between the curve and its asymptote be re- § VIII.] AREAS GENERATED BY VARIABLE LINES. 125 quired. We may regard this area as generated by the line PQ parallel to the axis of x, y being taken as the independent variable. Now PQ = 2a — Xy hence the required area is A = {2a — x) dy . . . J - CO (2) From the equation (i) of the curve, we have whence 2a — X = y + 4«' ,2' and equation (2) becomes dy Fig. 4. A = 8«3 - ,-4«nan-'^ _„/+4«- 2a = 47ra- Oblique Coordinates. 105. When the coordinate axes are oblique, if a denotes the angle between them, and the ordinate is the generating line, the differential of its motion in a direction perpendicular to its length is evidently sin a-dx ; therefore, the expression for the area is A = sin a \y dx. 126 GEOMETRICAL APPLICATIONS. [Art. 105. As an illustration let the area between a parabola and a chord passing through the focus be required. It is shown in treatises on conic sections, the expression for a focal chord is yi^ = 4rtCosec^rt', . . . (i) where a is the inclination of the chord to the axis of the curve, and a is the distance from the focus to the vertex. It is also shown that the equation of the curve referred to the diameter which bisects the chord, and the tan- gent at its extremity which is parallel to the chord is y = 4^ cosec^ ci'X (2) The required area may be generated by the double ordi- nate in this equation; and since from (i) the final value of y \s -^ 2a cosec^ a, equation (2) gives for the final value of x OR = a cosec^ a. Hence we have ta cosec'a ^ = 2 sin n y dx, J o or by equation (2) ' <=°^e<='" , 2>a^ cosec^ a ta cosec'a A = 4Va \ \/x dx = J o Employment of an Auxiliary Variable. 106. We have hitherto assumed that, in the expression A^^ydx, § VI 1 1 .] EMPLO YMENT OF AN A UXILIAR V VARIABLE. 1 27 X is taken as the independent variable, so that dx may be assumed constant ; and it is usual to take the limits in such a manner that dx is positive. The resulting value of A will then have the sign of y, and will change sign if y changes sign. It is frequently desirable, however, as in the illustration given below, to express both y and dx in terms of some other variable. When this is done, it is to be noticed that it is not necessary that dx should retain the same sign throughout the entire integral. The limits may often be so taken that the ex- tremityof the generating ordinate must pass completely around a closed curve, and in that case it is easily seen that the com- plete integral, which represents the algebraic sum of the areas generated positively and negatively, will be the whole area of the closed curve. (07. As an illustration, let the whole area of the closed curve -a) + \b = I, represented in Fig. 6, be required. If in this equation we put s.n ^', we shall have ^j =cos?/?; whence x = a sin^ -■/', and y = b cos^ '/? . (I) Therefore y dx — '^ab V cos^ '/.' sin^ rp dip. 128 GEOMETRICAL APPLICATIONS. [Art. 107. Now if in this integral we use the limits o and 2;r, the point determined by equation (i) de- scribes the whole curve in the direction A BCD A. Hence we have for the whole area A = ^ad cos* tp sin^ '/• dtl', and by formula (Q) A = lab i 2/T = ^— — . The areas in this case are all generated with the positive sign, since when y is negative dx is also negative. Had the generating point moved about the curve in the opposite direc- tion, the result would have been negative. Area generated by a Rotating Line or Radius Vector. 108. The radius vector of a curve given in polar coordinates is a variable line rotating about a fixed extremity. The angular rate is denoted by — and may be dt re- garded as constant, although the rate at which area is generated by the radius vector OP, Fig. 7, is not constant, be- cause the length of OP is not constant. The differential of this area is the area which would be generated in the time dt, if the rate of the area were con- stant ; that is to say, if the radius vector were of constant Fig. § VIII.] AREAS GENERATED BY ROTATING LINES. I29 length. It is therefore the circular sector O PR oi which, the radius is r and the angle at the centre is dd. Since arc PR^r dd, sector OPR = -r^ dd; therefore the expression for the generated area is A r" dd 109. As an illustration, let us find the area of the right-hand loop of the lemniscata 7^ ^ o? cos 2d. Fig. 8. (I) The limits to be employed are those values of d which make r = o ; that is and -, 4 4 Hence the area of the loop is A=—l' cos 2^ ^6*=: -sin 28 2]^^ 4 110. When the radii vectores, r« and ri corresponding to the same value of 6 in two curves, have the same sign, the area generated by their difference is the difference of the polar areas generated by 7\ and r^. Kence the expression for this area is A=~\{ri-r,') dd. (2) 130 GEOMETRICAL APPLICATIONS. [Art. III. III. Let us apply this formula to find the whole area between the cissoid rj = 2a (sec B — cos 6), Fig. 9, and its asymptote BP o and > a. — ^^ ~- • 32^8 9. Find the area of the loop of the curve ay = x*{b + x). 105^3 10. Find the area included between the axes and the curve ft -{if- t II. Ti n is an integer, prove that the area included between the axes and the curve ©""-(i) + i^r = . . n{n— 1) ' ' • 1 , IS A = — - — ^ r-^ — , . ab. 211 [2/1 — I) • ' ' [n + I) 12. If nis an odd integer, prove that the area included between the axes and the curve 0)"-(l)' = _ \fi{n— 2)" • ij Ttab IS ^ ^— 7 c • 2n {2n — 2} • • • 2 2 136 GEOMETRICAL AJPPLICATIONS. [Ex- VIII. 13. In the case of the curtate cycloid X = fl-'/' — b sin '/', y = a — b cos ?/', find the area between the axis of -v and the arc below this axis. (2a' + Z'') cos -'I - 2,^ Vib' - a'). 14. If ^ = ^art, show that the area of the loop of the curtate cycloid is 15. Find the area of the segment of the hyperbola -r = a sec ip, y = b tan ?/', cut off by the double ordinate whose length is 2b. ab V2 — log tan — 16. Find the whole area of the curve r' = a'' cos' + b" sin' 0. 17. Find the area of a loop of the curve r' = a' cos" — ^' sin' 0. - (a' + b'). 2 ab {a'-b') .,a — + ^^- -' tan -. 22 b 18. Find the areas of the large and of each of the small loops of the curve r ■= a cos cos 2O ; § VIII.] EXAMPLES. 137 and show that the sum of the loops may be expressed by a single integral. na^ _ «* . TTa" a" —T- + - , and . 16 4 ' 32 8 19. In the case of the spiral of Archimedes, r = aB, find the area generated by the radius vector of the first whorl and that generated by the difference between the radii vectores of the nth. and {u + i)th whorl. 6 20. Find the area of a loop of the curve and Sna n na 12 r = « sm Tfi, 21. Find the area of the cardioid r = 4rt! sin' 4^0. ^nc^. 22. Find the area of the loop of the curve cos 29 «" (4 — n\ r ^=^ a . - — ^-^^ -, cos (5 2 23. In the case of the hyperbolic spiral, r^ = a, show that the area generated by the radius vector is proportional to the difference between its initial and its final value. 138 GEOMETRICAL APPLICATIONS. [Ex. VIII. 24. Find the area of a loop of the curve r -=■ a cos ti 0. va 4« 25. Find the area of a loop of tlie curve 3 ., sin -5G a cos i 2 26. Find the area of a loop of the curve r sin = (7 cos 2'^ Notice that r /V r^a/ and finite from = ^— to^i =: — , and that -. — •^ -^ 4 4 ' J sm G iV negative in this interval. ^M 1^2 — log (i + Vs) . 27. Find the area of a loO]) of the curve (x- +fr^a\xy, a' Transform to polar coordinates. — . 28. In the case of the lima9on ;' = 2a cos 6r + <5, find the whole area of the curve when /' > 2a and show that the same expression gives the sum of the loops when b < 2a, {20" + lr)7t. 139 § VIII.] EXAMPLES. 29. Find separately the areas of the large and small loops of the limacon when b <.2a. \i a = COS"' ( — — large loop = {20" + b"") a + ^ ^(4^^ - I,') ; small loop = {20^ + b") [tt — a) — — ^(4^^ — i>'). 30. Find the area of a loop of the curve r^ = a^ cos nfi + b' sin 71 9. 31. Find the area of the loop of the curve 2 cos 26 — I r = a -7, , cos u V(a' + b*^ '^'-T. 32. Show that the sectorial area between the axis of -v, the equi- lateral hyperbola .r^-/=i, and the radius vector making the angle ^ at the centre is represented by the formula . I , I + tan /I =: - log ; 4 ^ I — tan fj ' and hence show that X = ■ , and y = . If A denotes the corresponding area m the case of the circle we have .r'+/=i, X = cos 2A, and y = sin 2 A. 140 GEOMETRICAL APrLICATIONS. [Ex. VII I. In accordance with the analogy thus presented^ the values of x and y given above are called the hyperbolic cosine and the hyperbolic sine of 2 A. Thus £2A I J-2A ^-2A ^2A =■ cosh (2^), = sinh (2^), 33. Find the area of the loop of the curve A"* — sai^y + 2ay^ = o. 34. Find the area of the" loop of the curve 38. Trace the curve . y x — 2a sin — , -V 35-^' ^■2« + i -f-y« + i — (2// + i) a.vy. a . 35. Find the area between the curve x^» + ^ jf.y^n+1 — (2,/ 4- i) ax"y" 2n + I J and its asymptote. ^^ 36. Find the area of the loop of the curve y 4- ax^ — axy = o. 37. Find the area of a loop of the curve X* + y* = a^xy. a 6^ ■na and find the area of one loop. nc^ §ix.] VOLUMES OF GEOMETRIC SOLIDS. 141 IX. Volumes of Geo7iietric Solids. 115. A geometric solid whose volume is required is fre- quently defined in such a way that the area of the plane sec- tion parallel to a fixed plane may be expressed in terms of the perpendicular distance of the section from the fixed plane. When this is the case, the solid is to be regarded as generated by the motion of the plane section, and its differential, when thus considered, is readily expressed. 116. For example, let us consider the solid whose surface is formed by the revolution of the curve APB, Fig. 11, about the axis OX. The plane section per- pendicular to the axis OX \s a circle; and if APB be referred to rectangu- lar coordinates, the distance of the section from a parallel plane passing through the origin is x, while the radius of the circle \s y. Supposing the centre of the section to move uniformly along the axis, the rate at which the volume is generated is not uniform, but its differential is the vol- ume which would be generated while the centre is describing the distance dx, if the rate were made constant. This differen- tial volume is therefore the cylinder whose altitude is dx, and the radius of whose base is j. Hence, if F denote the volume, dV = 7ty^ dx. 117. As an illustration, let it be required to find the volume of the paraboloid, whose height is h, and the radius of whose base is b. 142 GEOMETRICAL APPLICATIONS. [Art. 1 1 7. The revolving curve is in this case a parabola, whose equa- tion is of the form and since y = b when x = //, d^ = 4a/i, whence 4rz = y ; /i the equation of the parabola is therefore Hence the volume required is y= Tt \ y" dx 71 — - X dx — . //Jo 2 118. It can obviously be shown, by the method used in Art. 116, that whatever be the shape of the section parallel to a fixed plane, the differential of the volume is the product of the area of the generatitig section and the differential of its motion perpendicular to its pla?ie. If the volume is completely enclosed by a surface whose equation is given in the rectangular coordinates .r, j, s, and if we denote the areas of the sections perpendicular to the axes by Ajcy Ay, and A^, we may employ either of the formulas V= [A^dx, V= lAydv, V= [A.ds. The equation of the section perpendicular to the axis of x is determined by regarding x as constant in the equation of the surface, and its area A^ is of course a function of x. § IX.] VOLUMES OF GEOMETRIC SOLIDS. 1 43 For example, the equation of the surface of an ellipsoid is ■ h — + — = I The section perpendicular to the axis of x is the ellipse jj/2 ^ ^ ^2 _ ,j,2 ^ "^ ? ~cF' ' . b c whose semi-axes are- ^/ic? — x^) and - V(a^ — x^. a ^ a ^ ' Since the area of an eUipse is the product of n and its semi- axes, a'' ' The limits for x are ±a, the values between which x must lie to make the ellipse possible. Hence a^ i-a 3 (19. The area A,^ can frequently be determined by the con- ditions of the problem without finding the equation of the surface. For example, let it be required to find the volume of the solid generated by so moving an ellipse with constant major axis, that its center shall describe the major axis of a fixed ellipse, to whose plane it is perpendicular, while the ex- tremities of its minor axis describe the fixed eUipse. Let the equation of the fixed ellipse be — -1- — =. I ^2 ^ b^ ' 144 GEOMETRICAL APPLICATIOXS. [Art. I 19. and let c be the major semi-axis of the moving ellipse. The minor semi-axis of this ellipse is y. Since the area of an ellipse is equal to n multiplied by the product of its semi-axes, we have a Ttbc (" Therefore V= — V{a^ — -r^)(^-v; U J —a hence, see formula (J/), V = i^abc The Solid of Revoliltio7i regarded as Generated by a Cylindrical Stirface. 120. A solid of revolution may be generated in another ji manner, which is sometimes more convenient than the employment of a circular section, as in Art. 1 16. For example, let the cissoid FOR, Fig. 12, whose equation is ..."•"■■^ y^ {2a — x) = x^, revolve about its asymptote AB. The line PR, parallel to AB and terminated by the curve, describes a cylindrical surface. If we con- ceive the radius of this cylinder to pass from the value GA = 2a to zero, the cylindrical surface will evidently generate the solid of revolution. Now every Ji. 1 \ A — ^ y .4 \ T" p ? / Fig. 12. IX.] DOUBLE INTEGRATION. 145 point of this cylindrical surface moves with a rate equal to that of the radius; therefore the differential of the solid is the product of the cylindrical surface, and the differential of the radius. The radius and altitude in this case are PC= ia — X, and PR = 2j, therefore V = 47T \ {2ax — x^)-x dx. Putting X — a = a sin 6, V= 47ra^ ' „(cos^ 6 + cos^ 6 sin 6) dB — 27rV. 2 Do2iblc l7itco;7'ation. 121. When rectangular coordinates are used, the expression for the area generated by a line parallel to the axis of y and terminated by two curves is ^ = (ja-Ji)^-^- (I) A C Fig. 13. Let AB, in Fig. 13, be the initial, and CD the final position of the gen- erating line, then the area is ABDC, which is enclosed by the curves and by the straight lines X = b. 146 GEOMETRICAL APPLICATIONS. [Art. 121. If in equation (i) we substitute for jj — Ji the equivalent ex- pression dy, we have A=\' \''dydx, (2) which expresses the area in the form of a double integral. In this double integral the limits j'l and _;'2 forj', are functions of x, while a and d, the limits for x, are constants. 122. If the area is that of a closed curve j'l and j'2 are two values of J corresponding to the same value of x in the equa- tion of the curve, and a and d are the values of x for which j'j and j2 become equal, as represented by the dotted lines in Fig. 13. It is evident that the entire area may also be expressed in the form A = ^yxdy; (3) and that when either of the forms (2) or (3) is applied to the area of a closed curve the limits are completely determined by the equation of the curve. 123. The limits in either of the expressions (i) or (2) define a certain closed boundary, and since either of these integrals represents the included area, it is evident that wc may write dj' dx = dx dy ; provided it is understood that the limits in the two expressions are such as to represent the same boundary. It should however be noticed that if the boundary is like that represented by the full lines in Fig. 13, or if the arcs y — y\ and y =yi do not belong to the sainc curve, we cannot make a practical application of the form (3) without breaking up t'he integral into several parts. IX.] DOUBLE INTEGRATION. 147 124, Let 4> {p-'iy) t>e any function of x and j/. In the double integral I f ' <}){x,y)dydx, (i) X is considered as a constant or independent of y in the first integration, but the limits of this integration are functions of x. The double integration is then said to extend over the area which is represented by the expression \' dydx, or [ {y^-yi)dx. J a J.i , J a (2) 125. Now let the surface, of which S~ (I) (.r, y) . (3) is the equation in rectangular coordinates, be constructed ; and let a cylindrical surface be formed by moving a line perpen- dicular to the plane of xj about the boundary of the area (2) over which the integration extends. Let us suppose the value of s to be positive for all values of x and y which represent points within this boundary. Then the cylindrical surface, together with the plane of xy and the surface (3), encloses a solid, of which the base is the area (2) in the plane xy, or ASBR in Fig. 14, and the upper surface is CQDP a portion of the surface (3). Let SRPQ be a section of this solid perpendicular to the axis of x. In this section x has a constant value, and the ordinates of R and 5 are the corresponding values of j'l and y^. ^ The area of this section, which denote Fig. 14. P D c ^-^ \y.. .■■■'" h X A L 148 GEOMETRICAL APPLICATIOXS. [Art. 1 25. by A.ry as in Art. 1 1 7, may be regarded as generated by the line z, hence hi and therefore V= I P% rt^;/ rt'.v, (l) which is identical with expression (i) Art. 124. 126- Now it is evident that the same volume may be ex- pressed by V= \\::dxdy, provided that the double integration exte?ids over the same area. Hence, with this understanding, we may write J U (-^' jO dydx = \\ {x, y) dx dy. In this formula x and y may be regarded as taking the places of any two variables, the limits of integration being determined by a given relation between the variables. Thus we may write (j) {li, t') dv du — (}> (u, v) du dv, provided the limits of integration are determined in each case by the same relation between u and v. 127. For example, if this relation is u^ + 1^ — e'^ = o, § IX.] DOUBLE INTEGRATION. 1 49 the range of values in the first integration is between that is, we must have v^ < c^ — u^, or 21^ + v'^ — c'^ < o (i) But this condition also expresses the limits for ti, since v is only possible when t^ < c^. Now, putting rectangular coordi- nates, X and J, in place of ti and v, it is convenient to express the restriction (i), by saying that the range of values of x and y is such as to represent every point zvit/mi the circle ;^ + y _ ^2 ^ o. Volumes by Dottble and Triple Integration. 128. As an application of formula (i), Art. 125, let us sup- pose the curve ASBR to be the circle {x-hf^-{v-kf = c\ (I) and the equation of the surface CQDP to be xy = pz (2) Then V = -\ xy dy dx = — \ {y^-yl)xdx, p Ja iy^ 2p J a 150 GEOMETRICAL APPLTCATIO.VS. [Art. 128. in which the limits j\ and j, are derived from equation (i). Hence and V=~ f " Vic" - {x - hf] X dx. P ]a The limits for ;ir are the extreme values of x which makej» possible ; that is, a =^ h — c and b = h -V c. To evaluate the integral, put X — h = c sin B ; then V^—[\cos^d{h + c sin d) dd. Since, by Art. 87, It V ^zo^Bsm Odd =0, we have finally r= rkhc^ 129. A volume in general may be represented by the triple integral F = dz dy dx, (I) § IX.] TRIPLE INTEGRATION. 15 1 which is equivalent to V=\\{z^-z^dydx^ (2) for (^2 ~ -i) (^ = Ax^ the area of a section perpendicular to the axis of x. We may regard this formula as expressing the difference between two cylindrical solids of the form represented in Fig. 14. 130. When the volume is that of a closed surface, z^ and z-^ are two values of z in terms of x and y found from the equa- tion of the surface. The area over which the integration extends is in this case the projection of the solid upon the plane of xy ; in other words, the base of a circumscribing cylinder. Thus, if the volume is that of the sphere x^+ f ^ {z-~cf=a^, (I) Zx and ^"2 are the two values of z derived from this equation • that is c± V(n- — x^—y^). Hence z^ — Zi = 2 V{a^ — x^ — f). and F= 2 [[ \' {a"- - x'-f) dydx. c . . . (2) The integration here extends over the circle x'+f-a^^Q (3) 152 GEOMETRICAL APPLICATIONS. [Art. 130. since -Co — c^ is real only when a^ — x^ — }'^ y o. From equation (3) we find the limits forjj/ to be hence, by formula {M), equation (2) becomes V= TlUd^ -x^)dx. Finally the limits for x are ± a, since y is real only when x is between these limits ; therefore V r I 1'' = 7t C?X X^ = L 3 J -a ^;r^. Elements of Area and Volume. 131. In accordance with Art. 100, the expression for an area, f \\iydx (i) is the limit of the sum Since each of the terms included in 2^'' Aj is multiplied by the common factor A.r, this sum may be written in the form :^l:^j,>jA;p (2) § IX.] ELEMENTS OF AREA AND VOLUME. 1 53 The sum (2) consists of terms of the form and this product is called tJic clement of the sum ; in like man- ner, the product dy dx, which takes the place of Ay Ax when we pass to the limit by substituting integration for summation, is called the element of the integral (i), or of the area represented by it. 132. We may now regard the process of double integration as a process of double summation, as indicated by expression (2), followed by the act of passing to the limiting value. In the first summation indicated, the elemental rectangles corre- sponding to the same value of x are combined into the term (j2 — J'l) -^-^'j which may be called a linear element of area, since its length is independent of the symbol A . 133. It is easy to see that, in a similar manner, when rec- tangular coordinates are used, a volume may be regarded as the limiting value of the sum of terms of the form A X Ay A2\ and hence dx dy ds, which takes its place when we pass to the limiting value by substituting integration for summation, is called tJie element of volume. If the summation is effected in the order z, y, x, the first operation combines the elements which have common values oi y and x into the linear element of volume^ (^2- ^1) AX Ay. 154 GEOMETRICAL APPLICATIONS. [Art. I33, The second operation combines the linear elements correspond- ing to a common value of x, over a certain range of values oi y, into a term whose limiting value takes the form This last expression represents a lainina perpendicular to the axis of X, whose area is A^ a section of the solid, and whose thickness is a. v. Polar Elements. 134. If in the formula for a polar area, = l|(;V-;f)^^, (I) A [equation (2), Art. no], we substitute for -{r^— r^) the equi\ alent expression r dr, we obtain \drdff, (2) in which (x and ft are fixed limits for 6. Now it follows, from Art. 126, that the limits being deter- mined by a certain relation between r and 6, this integral may also be put in the form A=\\\''de.dr=\\{H^-e,)dr, ... (3) § IX.] POLAR ELEMENTS. 1 55 in which a and b are the limiting values of r, between which 6 is possible. The expression r dr dd, in equation (2), is called t\\Q polar element of area.* 135. The formula A^\^r{e^-e,)dr may also be derived geometrically ; for ;• (^2 — B^ is the length of an arc whose radius is r. As r increases, this arc generates the surface, and it is plain that every point has a motion, whose differential is dr, in a direction perpendicular to the arc. 136. In determining the volume of a solid, it is sometimes convenient to express ^ as a function of the polar coordinates of its projection in the plane of xy. In this case we employ the linear element of volume, (^2 - -1) r dr dO, corresponding to the polar element of area. * It is easily shown that the area induded between the circles whose radii are rand ;■ + Ar, and the radii whose inclinations to the initial line are 9 and 9 + A9 is (r + ^ A r) A r A 6. Since r+ iAr is intermediate between r and r + Ar, the limiting value of the sum, of which this is the element, is, by Art. 99, the integral of the element In the summation corresponding to equation (i), the elements are first combined into the sectorial element l(r-i _ r^) A6 ; while in the summation corresponding to equation (3), they are first combined into the arc-shaped element (r + 4Ar)(«2 — di) Ar. 156 GEOMETRICAL APPLTCATIONS. [Art. 1 36. As an illustration, let us determine the volume cut from a sphere by a right cylinder, having a radius of the sphere for one of its diameters. Taking the centre of the sphere as the origin, the diameter of the cylinder as initial line, and the axis of z parallel to the axis of the cylinder, we have for every point on the surface of the sphere ^ + r'' = a\ (I) where a is the radius of the sphere. Hence and V= 2\['\a^~r^)Kdrdft = [\'- -(«2_^^i dd. The circular base passes through the pole, and its equation is r = a cos 0, (2) hence the limits for r are o and a cos 6, and by substitution we obtain 2/7^ f r=— (I -sin^^)^^. The limits for B are ± - , the values which make r vanish 2 in equation (2) ; but it is to be noticed that the expression (^ — r^)8, for which we have substituted ci? sin^ 6, is always posi- tive, whereas sin^ d is negative in the fourth quadrant. Hence the value of V is double the value of the integral in the first quadrant ; that is, V= — (i — sm' 8) dd = . 3 Jo ^ 3 9 §IX.] POLAR ELEMENTS. 157 If a second cylinder whose diameter is the opposite radius of the sphere be constructed, the whole volume removed from the sphere is , and the portion of the sphere which remains is , a quantity commensurable with the cube of the diameter. Polar Coordinates in Space. 137. A point in space may be determined by the polar coordinates p, ^, and ^, of which p de- notes the radius vector OP, Fig. 15, 4) the inclination POR of /a to a fixed plane passing through the pole, and d the angle ROA, which the projection of p upon this plane makes with a fixed line in the plane. The angles (}) and 9 thus correspond to the latitude and longitude of the point P considered as situated upon the surface of a sphere whose radius is p. The radius of the circle of latitude BP is -2 Fig. 15. PC = p cos ^. The motions of P, when p, (f>, and 6 independently vary, are in the directions of the radius vector OP and of the tangents at Pto the arcs PR and PB. The differentials of these motions are respectively dp, pd^, and p cos ^dO; 158 GEOMETRICAL APPLICATIOXS. [Art. 1 37. and since these motions are mutually rectangular, the element of volume is their product, f? cos ^ dp d^ dd, and V =\\ (^ cos d(l> do (2) When the radius vector lies entirely within the solid, the lower limit px must be taken equal zero, and we may write V = -\\p'' cos (j) d(l> dd (3) The element of this double integral has the form of a pyramid with vertex at the pole. If, on the other hand, in formula (i) we perform first the integration with respect to (f>, we have V = (sin ^ — sin (f>^ p- dp dO. (4) Taking the lower limit d^ ; (6) and in each of the formulas the factor 2n may take the place of the integration with reference to d. 140. As an example of the use of equation (6), let us find the volume generated by a circle revolving about one of its tangents. The initial line, being perpendicular to the axis of revolution, is a diameter; hence if a is the radius of the circle its equation is p ^= 2a cos (f), 7t 7t and the limits for (j) are and — . Substituting in (6) V = iGna^ co?,*(j) d^ — 2n^a^. 141. The following example of the use of equation (4), Art. 138, is added to illustrate the necessity of drawing a figure in each case to determine the limits to be employed. Let it be required to find the volume generated by the revolution of the cardioid about its axis, the equation of the curve being p — a{i + sin^), . . Fig. 16. (1) l6o GEOMETRICAL APPLICATIONS. [Art. I4I. when the initial line is perpendicular to the axis of the curve, as in Fig. 16. The figure shows that the upper limit for ^ is - n, while the lower limit is the value of ^ given by equa- tion (1) ; therefore sm02=i, and sm pi = 1. The limits for p are evidently o and 2a. Substituting in equa- tion (4) Art. 138, = 27rr— -^'1" = ^-^— L 3 4«-io 3 Examples IX. I. Find the volume of the spheroid produced by the revolution of the ellipse, a- about the axis of x. ~ . 2. Find the volume of a right cone whose altitude is a, and the radius of whose base is b. rtaP 3. Find the volume of the solid produced by the revolution about the axis of .v of the area between this axis, the cissoid j'« {2a - x) = .v', and the ordinate of the point (a, a). 8aV (log 2 — |). IX.] EXAMPLES. l6l 4. Find the volume generated by the revolution of the witch, y^x — 2ay^ + /^a^x = o, 2_3 about its asymptote. See Art. 104. 47r*^ 5. The equilateral hyperbola X — y =i at revolves about the axis of x : show that the volume cut off by a plane cutting the axis of .r perpendicularly at a distance a from the vertex is equal to a sphere whose radius is a. 6. An anchor ring is formed by the revolution of a circle whose radius is l> about a straight line in its plane at a distance a froni its centre : find its volume. 27t^ad\ 7. Express the volume of a segment of a sphere in terms of the altitude /i and the radii Ui and Uc, of the bases. ~ {h' + za^ + 3a./). 8. Find the volume generated by the revolution of the cycloid, A" = fl! {ip — sin ^'), y :=z a {i — cos if:), about its base. 57rV. 9. The area included between the cycloid and tangents at the cusp and at the vertex revolves about the latter ; find the volume gen- erated. n a . 10. Find the volume generated by the revolution of the part of the curve y^E-, which is on the left of the origin, about the axis of x. n 2 * 1 62 GEOMETRICAL APPLICATIONS. [Ex. IX II. The axes of two equal right circular cylinders, whose common radius is a, intersect at the angle a ; find the volume common to the cylinders. T/te section Parallel to the axes is a rhombus. i6' and the surfaces of the paraboloid and of the cylinder. Z'^a* 34. Find the volume cut from a sphere whose radius is a by a right circular cylinder whose radius is i, and whose axis passes through the centre of the sphere. 47r 3 {a'-n'']. 35. Find the volume cut from a sphere whose radius is a by the cylinder whose base is the curve r — a cos 30. 2a^7t Sa' 3 9 36. Find the volume cut from a sphere whose radius is a by the cylinder whose base is the curve r = a cos + sm 0, 7 ^ 47r^^ 16 , J ^3 supposmg b b, about the axis oi y. Transform to polar coordinates, and use the method of Art. 139. 6 ^ 2'^{a'-b'y a' 43. Find the volume generated by the curve given in the preceding example, when revolving about the axis of x. Tta {2a' + 3b') Tib" a + \/{a'' - i') 1 68 GEOMETRICAL APPLICATIONS. [Ex. IX. 44. Find the volume common to the sphere whose radius is p = a, and to the solid formed by the revolution of the cardioid, r = a (i 4- cos O), about the initial line. See Art. 141. —r~- 45. Find the whole volume enclosed by the surface Transform to the coordinates p, (/>, 0, and show that the solid consists a' 0/ four equal detcuhed parts. —. X. Rectificatio7i of Plajic Curves. 142. A curve is said to be rectified when its length is deter- mined, the unit of measure to which it is referred being a right line. It is shown in Diff. Calc, Art. 314 [Abridged Ed., Art. 164], that, if s denotes the length of the arc of a curve given in rectangular coordinates, we shall have ds = V{d.x^ + df). If the abscissas of the extremities of the arc are known, s is found by substituting for dy in this expression its value in terms of x and dx, and integrating the result between the given values of x as limits. Thus, to express the arc measured from the vertex of the semi-cubical parabola aji^= x^ § X.] RECTIFICATION OF PLANE CURVES. l6g in terms of the abscissa of its other extremity, we derive, from the equation of the curve, dy — 3 Vxdx 2Va ' whence 2 Va Integrating, s = \/(gx + 4a) dx 2 |/rt Jo I , .a %a = — {gx + 4^)' . 27 Va 27 143. When x and y are given in terms of a third variable, ds is generally expressed in terms of this variable. For exam- ple, from the equations of the four-cusped hypocycloid, X — a cos^ ^', y — a sin^ ip, . . . (i) we derive dx = — 3^ cos^ tp sin ip dip, and dy = 2^ sin^ ip cos ip dip; whence ds = T^a sin ip cos tp dtp (2) The length of the arc between the point {a, 6), corresponding to ip = o, and (o, a) corresponding to ip = | tt, is therefore 3^sins/ 2 3f 2 I/O GEOMETRICAL APPLICATIONS. [Art. 1 44. Change of tJic Sign of ds. (44. Wc have hitherto assumed ds to be positive, but it is to be remarked that an expression substituted for ds^ as in the illustration given in the preceding article, may change sign. Thus, in equation (2), ds, which is so written as to be positive while //• passes from o to i/T, becomes negative while ^- passes from \7r to n. Thus the integral gives a negative result for the arc between the points (o, a) and {—a, o), corresponding to \n and n. This change of sign in ds indicates a cusp or sta- tionary point of the curve ; and the existence of such points must be considered before we can properly interpret the result- ing values of s. For instance, if in this example we integrate between the limits o and — , we get the results = -- , which is 4 4 the algebraic sum, but the numerical difference of the arcs between the points corresponding to the limits. Polar Coordinates. 145. It is proved in Diff. Calc, Art. 317 [Abridged Ed., Art. 167], that when the curve is given in polar coordinates ds = V(dr^ + r^ d6^). This is usually expressed in terms of B. For example, the equation of the cardioid is r — a {i — cos B) z= 2a sin^ 4 ; whence dr = 2a sin -jB cos ^0 d6, and by substitution ds = 2a sin | /V dB. § X.] RECTIFICATION OF CURVES. I7I The limits for the whole perimeter of the curve are o and 27r, and ds remains positive for the whole interval. Therefore s =■ 2a\ sin —dd^= —4a cos Jo 2 = 8^. Rectification of Curves of Dotcble C^crvattcre. 146. Let ff denote the length of the arc of a curve of double curvature ; that is, one which does not lie in a plane, and sup- pose the curve to be referred to rectangular coordinates ;f, j^ and ^'. If at any point of the curve the differentials of the coordinates be drawn in the directions of their respective axes, a rectangular parallelopiped will be formed, whose sides are dxydy and ds^ and whose diagonal is da. Hence da = V{dx^ + df + d:?). The curve is determined by means of two equations connect- ing .r, y and z, one of which usually expresses the value of y in terms of x, and the other that of z in terms of x. We can then express d(S in terms of x and dx. If the given equations contain all the variables, equations of the required form may be obtained by elimination. 147. An equation containing the two variables x and y only is evidently the equation of tJie projection upon the plane of xy of a curve traced upon the surface determined by the other equation. Let j- denote the length of this projection; then, since ds^ = dx^ + df', d^ = V{ds^ + d^), in which ds may, if convenient, be expressed in polar coordin- ates ; thus, d(j= V{dr^ "^ r^dS'^ ^ dz% 172 GEOMETRICAL APPLICATIONS. [Art. 1 48. !48. As an illustration, let us use this formula to deter- mine the length of the loxodromic curve from the equation of the sphere, x" ^ f ^ :? = a', (I) upon which it is traced, and its projection upon the plane of the equator, of which the equation is 2a - ^{x^ + /) ii." '''>"-'!^ + f-« '^" ij , or in polar coordinates 2a = r («"" + f-'-") (2) Equation (i) is equivalent to r= + ^ = ^2 ; and, denoting the latitude of the projected point by ^, this gives ^ = rt' sin ^, r = a cos ^. . . . (3) In order to express dQ in terms of ^, we substitute the value of r in (2) ; whence f «e J. f - «e — 2 sec ^, (4) and by differentiation £"" - f-«^ = - sec ^ tan c5 -J? (5) ;/ dO Squaring and subtracting equation (5) from equation (4), 4 sec^^r 2 iA^^~\ 4-^^L.^-tan^^^-^J, which reduces to de'' = ^^l±^ (6) § X,] LENGTH OF THE LOXODROMTC CURVE. 1/3 From equations (3) and (6) dz^ = cr cos^ ^ d<^ ; whence substituting in the value of dG (p. 171) d6=:av{i+^d^. Integrating, G = a—^ '\ d

= o and tp = rr ; and find the whole length of the curve. 12a. 10. Find the length of a quadrant of the curve r r- c A * d' + ad + F See Fis". 6, Art. 107. — . 11. Show that the curve X ^=^ 2a cos''' ^ (3 — 2 cos" 6), y = ^a sin cos' (9 has three cusps, and that the length of each branch is • — . 3 12. Find the length of the arc between the points at which the curve X = a cos'' cos 26, _^ = « sin ^ ^ sin 2 ^ 2 — V2 cuts the axes. a. 176 GEOMETRICAL APPLICATIONS. [Ex. X. 13. Show that the curve -v = a cos ^" (i + sin'' a -\r b , x= (a -V h) cos ip — cos — - — ?/', y ^=. {a -\- b) sm f — b sm — 7— ip. U {a + b) 15. Show that the curve .V = 9<2 sin y,' — 4a sin' '/•, y ^ — T^a cos '/• + 4a cos' '/• is symmetrical to the axes, and has double points and cusps : find the lengths of the arcs, (a) between the double points, (/S) between a double point and a cusp, and {y^ the arc connecting two cusps, and not passing through the double points. («, T^ 16. Find the whole length of the curve .V = 3. 2. Prove that the surface generated by the arc of the catenary given in Ex. X., I, revolving about the axis of .v, is equal to 7r{cx + J}'). 3. Find the whole surface of the oblate spheroid produced by the 1 84 GEOMETRICAL APPLICATIOXS. [Ex. XI. revolution of an ellipse about its minor axis, a denoting the major, b the mmor semi-axis, and e the excentncity, . . . b\ I +^ 2na + TT - log . e 1 — e 4. Find the whole surface of the prolate spheroid produced by the revolution of the ellipse about its major axis, using the same notation as in Ex. 3. , iSin-V 2 7ti? + 271 ab . 5. Find the surface generated by the cycloid X = a {ip — sin '/), y =^ a {i — cos tp) revolving about its base. — tco'. 3 6. Find the surface generated when the cycloid revolves about the tangent at its vertex. 3 7. Find the surface generated when the cycloid revolves about its \7Ta'{ Tt — - 8. Find the surface generated by the revolution of one branch of the tractrix (see Ex. X., 5) about its asymptote. 2 /Ta^ § XL] EXAMPLES. 185 9. Find the surface generated by the revolution about the axis of X of the portion of the curve y = f ", which is on the left of the axis of j'. 7r[Y2 + log (i + 12)]. 10. Find the surface generated by the revolution about the axis of -v of the arc between the points for which x = a and x — i> in the hyperbola xy = k\ 7tk' ' P + \'{k' + //) ^/{k' + a') _ six + b') II. Show that the surface of a cylinder whose generating lines are parallel to the axis of z is represented by the integral 6" = \z iis , where s denotes the arc of the base in the plane of xy. Hence, deduce the surface cut from a right circular cylinder whose radius is a, by a plane passing through the centre and making the angle (v with the plane of the base. 20^ tan a. 12. Find the surface of that portion of the cylinder in the problem solved in Art. 154, which is within the hemisphere. 2«^ 13. Find the surface of a circular spindle, a being the radius and 2c the chord. ^7ta c — s/ia — ^')sin-'- a 1 86 GEOMETRICAL APPLICATIONS. [Art. 1 5 5. XII. TJic Area generated by a Straight Line moving in any Manner i)i a Plane. 155. If a straight line of indefinite length moves in any man- ner whatever in a plane, there is at each instant a point of the line about which it may be regarded as rotating. This point we shall call the centre of rotation for the instant. The rate of motion of every point of the line in a direction perpendic- ular to the line itself is at the instant the same as it would be if the line were rotating at the same angular rate about this point as a fixed centre.* Hence it follows that the area generated by a definite portion of the line has at the instant the same rate as if the line were rotating about a fixed instead of a variable centre. (56. Suppose at first that the centre of rotation is on the generating line produced, pj and p^ denoting the distances from the centre of the extremities of the generating line, and let ^ denote its inclination to a fixed line. By substitution in the general formula derived in Art. no, we have dA = ^- {pi- pi) d(j>. * Compare Diff. Calc, Art. 332 [Abridged Ed., Art. 176J, where the moving linfe is the normal to a given cur\e, and the centre of rotation is the centre of cur- vature of the given curve. If the line is moving without change of direction, the centre is of course at an infinite distance. Wiien the line is regarded as forming a part of a rigidly connected system in motion, its centre of rotation is the foot of a perpendicular dropped upon it from the insUxutaueous centre of the motion of the system. Thus, if the tangent and normal in the illustration cited are rigidly connected, the centre of curvature, C, is the instantaneous centre of the motion of the system, and the point of contact, P, is the centre of rotation for the tangent line. XII.] AREAS GENERATED BY MOVING LINES. 1 8/ Applications. 157. The area between a curve and its evolute may be generated by the radius of curvature p, whose incHnation to the axis of ;ir is ^ + \7t, in which ^ denotes the inclination of the tangent line. Since the centre of rotation is one extremity of the generating line/?, the differential of this area is found by substituting in the general expression i\ = o and /32 = o. Hence when p is expressed in terms of ^, A = - \/rd(^ expresses the area between an arc of a given curve, its evolute, and the radii of curvature of its extremities, the limits being the values of (/> at the ends of the given arc. 158. For example, in the case of the cardioid r = a{i — cos 6), it is readily shown, from the results obtained in Art. 145, that the angle between the tangent and the radius vector is ^6^; and therefore ^ = f ^, and ds Aa . 6 d 3 3 To obtain the whole area between the curve and its evolute, the limits for are O and 2/T ; hence the limits for ^ are o and 37r. Therefore 2 8^2 f? d(l) 9 (f) , , ^7td sin^ ~ d(l) = 3 159. As another application of the general formula of Art. 156, let one end of a line of fixed length a be moved 1 88 GEOMETRICAL APPLICATIONS. [Art. 1 59. along a given line in a horizontal plane, while a weight at- tached to the other extremity is drawn over the plane by the line, and is therefore always moving in the direction of the line itself. The line of fixed length in this case turns about the weight as a moving centre of rotation. Hence the area generated while the line turns through a given angle is the same as that of the corresponding sector of a circle whose radius is a. The curve described b)' the weight is called a tractrix, and the line along which the other extremity is moved is tJie direc- trix. When the axis of x is the directrix, and the weight starts from the point (o, a), the common tractrix is described ; hence the area between this curve and the axis is \Ttc?. 160. Again, in the generation of the cycloid, Diff. Calc, Art. 288 [Abridged Ed., Art. 156], the variable chord RP may be regarded as generating the area. The point R has a motion in the direction of the tangent RX \ the point /'partakes of this motion, which is the motion of the centre C, and also has an equal motion, due to the rotation of the circle in the direc- tion of the tangent to the circle at P. Since the tangents at P and R are equally inclined to PR, the motion of P in a direction perpendicular to PR is double the component, in this direction, of the motion of R. Therefore the centre of rota- tion of PR is beyond v'? at a distance from it equal to PR. Hence, denoting PRO by <5, pj — PR — 2a sin ^, />2 = 2PR = 4a sin ^. Substituting in the formula of Art. 156, we have for the area of the cycloid, since PRO varies from o to n. A = = 6a^ sin^

, (I) for the differential of the area, was obtained upon the supposi- tion that A and B were on the same side of C. Then suppos- ing P2 > Px, and that the line rotates in the positive direction^ as in figure 19, the differential of the area is positive; and we notice that every point in the area generated is swept over by the line AB, the left hand side as we face in the direction A B preceding. 162. We shall now show that in every case, the formula requires that an area swept over with the left side preceding, shall be considered as positively generated, and one swept over in the opposite direction as negatively generated. In the first place, if C is between A and B so that Pi is negative, as in figure 20, pi is still positive, and formula (i) still gives the difference between the areas generated by ^;5 and AC. Hence the latter area, which is now generated by a part of the line AB, must be regarded as generated negatively, but the right hand side as we face in the direction AB of this portion of the line is nov/ preceding, which agrees with the rule given in Art. 161. Again, if C is beyond B, the formula gives the difference of the generated areas; but since pi is numerically greater than pI, in this case, dA is negative, and the area generated by AB is the difference of the areas, and is negative by the rule. Fig. 20. 190 GEOMETRICAL APPLICATIONS. [Art. 162. Finally, if the direction of rotation be reversed, d^ and therefore dA change sign, but the opposite side of each por- tion of the line becomes in this case the preceding side. 163. We may now put the expression for the area in another form. For dA=- (p^ 2 ^ ' Pi) i^4> = (P2 ^,)a±Arf^: whatever be the signs of p^ and Pj, the first factor is the length of AB, which we shall denote by /, and the second factor is the distance of the middle point of AB from the centre of rotation, which we shall denote by p^. Hence, putting Ih - Pi = I, we have and A = Upmd^. Pl±Jh- n 2 ~^'"' (2) Since p,„d(p is the differential of the motion of the middle point in a direction perpendicular to AB, this expression shows that the differential of the area is the product of this differential by the length of the generating line. Areas generated by Lines whose Extremities describe Closed Circtiits. i^* 164. Let us now suppose the generating line AB to move from a given position, and to return to the same position, each of the extremities A and B describing a closed curve in the positive direction, as indicated by the arrows in figure 21. It is readily seen that every point which is in the area described by B, and not in that described by A, will be swept over at least once by the line AB, the left side preceding, Fig. 21. and if passed over more than once, there will be § XII.] AREAS GENERATED BY MOVING LINES. I9I an excess of one passage, the left side preceding. Therefore the area within the curve described by B, and not within that described by A, will be generated positively. In like manner the area within the curve described by A, and not within that described by i?, will be generated negatively. Furthermore, all points within both or neither of these curves are passed over, if at all, an equal number of times in each direction, so that the area common to the two curves and exterior to both disap- pears from the expression for the area generated by AB. Hence it follows that, regarding a closed area zuJiose perimeter is described in the positive direction as positive, the area generated by a line returning to its original position is the differe?tce of the areas described by its extremities. This theorem is evidently true generally, if areas described in the opposite direction are regarded as negative. Amslers Planimeter. 165. The theorem established in the preceding article may be used to demonstrate the correctness of the method by which an area is measured by means of the Polar Planimeter, invented by Professor Amsler, of Schaffhausen. This instrument consists of two bars, OA and AB, Fig. 22, jointed together at A. The rod OA turns on a fixed pivot at O, while a tracer at B is carried in the positive direction completely around the perimeter of the area to be measured. At some point C of the bar AB a small wheel is fixed, having its axis parallel to AB, and its circumference resting upon the paper. When ^is moved, this wheel has a sliding and a roll- ing motion ; the latter motion is recorded by an attachment by means of which the number Fig. 22. of turns and parts of a turn of the wheel are registered. 192 GEOMETRICAL APPLICATIONS. [Art. 1 66. 166. Let J/ be the middle point of AB, and let OAr^a, AB = b, MC = c. Since b is constant, the area described by yJ/> is by equation (2), Art. 163, Area^i5 = ^|p,„^/^ (i) Denoting the linear distance registered on the circumference of the wheel by s, ds is the differential of the motion of the point C, in a direction perpendicular to AB, and since the dis- tance of this point from the centre of rotation is f>„i + c, ds = {p„, + c) d^: substituting in (i) the value of p„,d(f>, - bc[d(}> (2) Area AB — b ds 167. Two cases arise in the use of the instrument. When, as represented in Fig. 22, O is outside the area to be meas- ured, the point A describes no area, and by the theorem of Art. 164, equation (2) represents simply the area described by B. In this case ^ returns to its original value, hence d(!> vanishes, and denoting the area to be measured by^, equation (2) becomes A = bs (3) In the second case, when O is within the curve traced by B, the point A describes a circle whose area is nd^, and the limit- § XII.] AMSLER'S PLANIMETER. 193 ing values of ^ differ by a complete revolution. Hence in this case equation (2) becomes A — Ttc? ^^ bs — 2 Tcbc, or A = bs + 7t{a^ — 2bc).'^ (4) In another form of the planimeter the point A moves in a straight line, and the same demonstration shows that the area is always equal to bs. Examples XII. I. The involute of a circle whose radius is a is drawn, and a tangent is drawn at the opposite end of the diameter which passes through the cusp ; find the area between the tangent and the involute. a'n (3 + n"-) 2. Two radii vectoresof a closed oval are drawn from a fixed point within, one of which is parallel to the tangent at the extremity of the other ; if the parallelogram be completed, the area of the locus of its vertex is double the area of the given oval. 3. Show that the area of the locus of the middle point of the chord joining the extremities of the radii vectores in Ex. 2, is one half the area of the given oval. * The planimeter is usually so constructed that the positive direction of rotation is with the hands of a watch. The bar b is adjustable, but the distance ^ C is fixed so that c varies with b. Denoting AChy q, we have c =^ q — \b, and the constant to be added becomes C=-it {a- — 2bq + b-) in which a and q are fixed and b adjusta- ble. In some instruments q is negative. It is to be noticed that in the second case s may be negative ; the area is then the numerical difference between the constant and bs. 194 GEOMETRICAL APTLICATIOXS. [Ex. XII. 4. Prove that the difference of the perimeters of two parallel ovals, whose distance is b, is 2nh, and that the difference of their areas is the product of b and the half sum of their perimeters. 5. A lima'.on is formed by taking a fixed distance be on the radius vector from a point on the circumference of a circle whose radius is a ; sho-.v that the area generated by b when b> 2a \% the area of the lima- 9on diminished by twice the area of the circle, and thence determine the area of the lima9on. 7r(2': + 15 (73 + ^'s) - 6 ( ji + je) - 2.oy,\ 20O GEOMETRICAL APPLICATIOXS. [Art. 1/2. and substituting 173. It is, however, to be noticed, that when the ordinates are symmetrically situated with respect to the area, if n is even, the parabolic curve may be assumed of the {n + i)th degree. For example, in Art. i68, n = 2, but the curve was assumed of the third degree. Inasmuch as A, B, C and D cannot all be expressed in terms of ji, y-i, and_)/3, we see that a variety of parabolic curves of the third degree can be passed through the extremities of the measured ordinates, but all of these curves have the same area.* Application to Solids. 174. \{ y denotes the area of the section of a solid perpen- dicular to the axis of -r, the volume of the solid is lydx, and * This circumstance indicates a probable advantage in making « an even num- ber when repeated applications of the rules are made. Thus, in the case of six intervals, we can make three applications of Simpson's first rule, giving A = —[i, 4, 2, 4, 2, 4, i], (I) or two of Simpson's second rule, giving ^ = ^ [i. 3, 3- 2, 3, 3, i] (2) In the first case, we assume the curve to consist of three arcs of the third degree, meeting at the extremities of the ordinates _j',j and jr. ; but, since each of these arcs contains an undetermined constant, we can assume them to have common tangents at tl^ points of meeting. We have therefore a smooth, though not a continuous curve. In the second case, we have two arcs of the third degree containing no arbitrary constants, and therefore making an angle at the extremity of y^. It is probable, therefore, that the smooth curve of the first case will in most cases form a better approximation than the broken curve of the second case. In confirmation of this conclusion, it will be noticed that the ratios of the coefficients in equation (i) are nearer to those of Cotes' coefficients for« = 6, given in the preceding foot-note, than are those in equation (2). §XIII.] APPLICATION TO SOLIDS. 201 therefore the approximate rules deduced in the preceding arti- cles apply to solids as well as to areas. Indeed, they may be applied to the approximate computation of any integral, by putting J equal to the coefficient of x under the integral sign. The areas of the sections may of course be computed by the approximate rules. Woolleys Rule. 175. When the base of the solid is rectangular, and the ordinates of the sections necessary to the application of Simp- son's first rule are measured, we may, instead of applying that rule, introduce the ordinates directly into the expression for the area in the following manner. Taking the plane of the base for the plane of xy, and its centre for the origin, let the equation of the upper surface be assumed of the form z=A ^Bx-^ Cy + D-x" + Exy + Ff + G.x^ + Hx'^y + Ixf + Jf. Let 2h and 2k be the dimensions of the base, and denote the measured values of r as indicated in Fig. 25. The required volume is '^ V = a dy dx. This double integral vanishes for every term containing an odd power of x or an odd power of J : hence ^,, \DJi^k ^FhB V= AAhk + + ; hk = —\\2A ^ aDJi^ ^- aFL^I (I) 3 202 GEOMETRICAL APPLICA TIOjVS. [Art. 1 75. By substituting the values of x and y in the equation of the surface, we readily obtain b2 = A, (2) «i + ^3 + ^1 + ^3 = 4^^ + A^^i^ + AF^, • • • (3) ^2 + ^j + Ih + h = 4A + 2D}? + 2FB. ... (4) From these equations two very simple expressions for the volume may be derived ; for, employing (2) and (4), equation (i) becomes 2hk F=~-(^2 + <5i + 2^, + <5'3+^2); . . . . (4) and employing (2) and (3), hk F= — (^1 + ^3 + 8^2 + ^1 + ^3) (5) Equation (4) is known as WoolUys Rule ; the ordinates employed are those at the middles of the sides and at the centre ; in (5), they are at the corners and at the centre. Examples XIII. 1. Apply Simpson's Rule to the sphere, the hemisphere, and the cone, and explain why the results are perfectly accurate. 2. Apply Simpson's Second Rule to the larger segment of a sphere made by a plane bisecting at right angles a radius of the sphere. ~8~" § XIIL] EXAMPLES. 203 3. Find by Simpson's Rule the volume of a segment of a sphere, b and c being the radii of the bases, and h the altitude. ?-(3'^= + 3^'^+n 4. Find by Simpson's Rule the volume of the frustum of a cone, b and c being the radii of the bases, and h the altitude. — (b'^ + be + c'). 3 5. Compute by Simpson's First and Second Rules, the value of , the common interval being -^^ in each case. The first rule gives 0.6931487, and the second rule gives 0.6931505. The correct value is obviously loge2 = 0.6931472. 6. Find the volume considered in Art. 175, directly by Simpson's Rule, and show that the result is consistent with equations (4) and (5). V= — [ai + as + Ci + C3 + 4 {a.2 + bi +b3 + c^) + lOb^]. 7. Find, by elimination, from equations (4) and (5), Art. 175, a formula which can be used when the centre ordinate is unknown. F= — [4{a, + b, + b, + c,) - (^1 +a, + ^1 + c,)]. o 204 MECHANICAL APPLICATIONS. [Art. 1 76. CHAPTER IV. Mechanical Applications. XIV. Definitions. 176. We shall give in this chapter a few of the applications of the Integral Calculus to mechanical questions. The iuass or quantity of matter contained in a body is pro- portional to its weight. When the masses of all parts of equal v^olume are equal, the body is said to be Jioviogcncoiis. The factor by which it is necessary to multiply the unit of volume to produce the unit of mass is called the density, and usually denoted by y. In the following articles it will be assumed, when not other- wise stated, that the body is homogeneous, and that the density is equal to unity, so that the unit of mass is identical with the unit of volume. When the mass of an area is spoken of, it is regarded as a lamina of uniform thickness and density, and the unit of mass is taken to correspond with the unit of surface. In like manner the unit of mass for a line is taken as identical with the unit of length. Statical Moments. VII. The viovicnt of a force, with reference to a point, is the measure of the effectiveness of the force in producing motion about the point. It is shown in treatises on Mechanics, that this is tlie proihict of the force and the perpetidicular from the point upon the line of application of the force. § XIV.] STATICAL MOMENTS. 205 The moment of the sum of a number of forces about a given point is the sum of the moments of the forces. The statical moDicnt of a body about a given point is the moment of its gravity ; the force of gravity being supposed to act upon every part of the body, and in parallel lines. 178. In order to find the statical moment of a continuous body, we regard the body as generated geometrically in some convenient manner, and determine the corresponding differen- tial of the moment. In the case of a plane area, let the body be referred to rectangular axes, and let gravity be supposed to act in the direction of the axis of j'. Then the abscissa of the point of application is the arm of the force when we consider the moment about the origin. Let us first suppose the area to be generated by the motion of the ordinate/. The differential of the area is then y dx. The corresponding element of the sum, of which the integral y dx is the limiting value, see Art. 99, is JVA.r, (i) in which jjv is the ordinate corresponding to any value of x intermediate between a + (r — i) A-r, and a + r :\x. It is evident that the arm of the weight of the element (i) is such an intermediate value of x ; hence the moment of the ele- ment is x^yr t\x (2) The whole moment is therefore the limiting value of a sum^ of the form ^ Xryr A;ir. In other words, it is the integral xydx, (3) 206 MECHANICAL APPLICA TIONS. [Art. 178. in which the differential of the moment is the product of the differential of the area and the arm of the force, which in this case is the same for every point of the clement. In other words, the inovient of the differential is the differential of the vtoment. 179. As an illustration, we find the moment of a semicircle (Fig. 26) about its centre. The area may be generated by the line 2r, moving from ;tr = o to Y Fig. 26. X = a. The equation of the circle being x" ■\- f = d\ the differential of the area is 2 4/(^2 _ ^^) ^^_ The moment of this differential is 2 V{a^ — x^)x dx ; hence the whole moment is 2f" \'{c^ ~ x^)xdx = - ? {cC- -^^fX ='^^, Centres of Gravity. 180. If a force equal to the whole weight of a body be applied with an arm properly determined, its moment may be made equivalent to the whole statical moment of the body. If the force is in the direction of the axis of;', as in Fig. 26, we have, denoting this arm by "x, Ic • Area = Moment, _ Moment ^ ~ Area ' § XIV.] CEA'TEES OF GRAVITY. 20/ In like manner, supposing the force to act in the direction of the axis of x, we may determine/ for the same body. It is shown in treatises on Mechanics that the point deter- mined by the two coordinates x and y, is independent of the position of the coordinate axis. This point is called the centre of gravity of the area. The centre of gravity of a volume is defined in like manner. 181. The symmetry of the form of a body may determine one oi more of the coordinates of its centre of gravity. Thus the centre of gravity of a circle or a sphere coincides with the geometrical centre, and the centre of gravity of a solid of revolu- tion is on the axis of revolution. The centre of gravity of the semicircle in Fig. 26, is on the axis of x ; hence to determine its position we have only to find x. Dividing the moment of the semicircle found in Art. 179 by the area \ncB., we have _ Aa X r=z — . (82. In finding the moment of the semicircle (Art. 179), we regarded the area as generated by the double ordinate 2y, and the differential of the moment was found by multiplying the differential of the area by x, which is the arm of the force for every point of the generating line. We may, however, derive the moment from the differential of area, ^'^J, (0 since the area may be generated by the motion of the abscissa X from J = — a to y = a. But in this case to find the moment of the differential we must multiply it by the distance of its centre of gravity from the given axis. The centre of gravity of the line x is evidently its middle point, hence the required ^rm is ^x. Therefore the differential of the moment is x^ dy , . -z^) (2) 20S MECHANICAL APPLICATIONS. [Art. 1 82. and consequently the whole moment is This result is identical with that derived in Art. 179. Polaj' Formulas, 183. When polar formulas are employed, r and B being coordinates of the curved boundary of the area, the element is \r^ dS. Since this element is ultimately a triangle, we employ the well known property of triangles ; that the centre of gravity is on a medial line at two-thirds the distance from the vertex to the base. The coordinates of the centre of gravity of the element are, therefore, 2 . 2 -rsin6' and —rcosO. 3 Hence we have the formula Urcosd^r^dO 2 [;^cos^^^ .*• = Ur'dt) ^1' fr^sin^rt'^ r^df^ 3 and similarly J' = ■ 3 jr^ de § XIV.] POLAR FORMULAS. 209 184. To illustrate, let us find the centre of gravity of the area enclosed by the lemniscata ^2 — ^j,2 ^Qg 26, (cos 2^) 'cos Odd Whence .r = = — 3 - 3 f'cos2^./(y (cos 2O) COS Odd. Put cos 26 — cos" 4>, whence sin (^ = ^2 sin d, and \'2 cos S dQ — cos ^ ^^, - 2i'2 f2 , , ,, 2\'2 l-l TT t 2 3 Jo "^ 3 4-2 2 b Solids of Revolution. ' 185. To find the centre of gravity of a solid of revolution, we take the axis of revolution as the axis of x, and the circle whose area is irf as the generating element. Replacing y in equation (3), Art. 178, by this expression, we have for the stati- cal moment 71 xjr dx, and for the abscissa of the centre of gravity ,rj'^ dx X — L"-, . y dx £• 210 MECHANICAL A P PLICA TIONS. [Art. 186. 186. To illustrate, wc find the centre of gravity of a spheri- cal segment whose height is li. In this case, taking the origin at the vertex of the segment, and denoting the radius of the sphere by a, we have ^1^ y ■2 ^. 2ax - ^. Hence {2ax' — x^) dx -ax^ - -x^ {2a X — x^)dx ax -X 3 J // Sa - ih 4. Vi — h ' If the centre of gravity of the surface of the scgvient be re- quired, bince the differential of the surface is 27Ty ds, we easily obtain the general formula x = xy ds ' o -yjt ' lyds and, in this case the curve being a circle,/ ds = a dx ; hence, substituting, we have X = \h. The Properties of Pappus. 187. Let a solid be generated by the revolution of any plane figure about an exterior axis in its own plane. It is required to determine the volume and the surface thus generated. It is evident that this solid may also be generated by a variable circular ring w^hose centre moves along the axis of revolution ; denoting by y^ and y^ corresponding ordinates of XIV.] THE PROPERTIES OF PAPPUS. 211 the outer and inner circles respectively, the area of this ring is '^{yx — yi). Hence V^ n\{y^--yi) dx = 2;rpA±Zi(_j,^ -j,;)dx. But this integral is the statical moment of the given figure, since j'l — j'^ is the generating element of its area, and — — ^is the corresponding arm. Denoting the area of the figure by A, we may therefore write V=27ryA; that is, //le volume is tJie product of the area of the figure and the path described by its ccfitre of gravity. The surface {S) of this solid is, by Art. 149, S = 2n\yds =.27T dSf ii J denotes the ordinate of the centre of gravity of the arc s. Hence we have 5 = 2ny-arc ; that is, tJie surface is the product of the lengtli of the arc into the path described by the centre of gravity. These theorems are frequently called the properties of Gul- dinus ; they are, however, due to Pappus, who published them 1588. It is obvious that both theorems are true for any part of a revolution of the generating figure. 212 MECHAXICAL APPLICATIONS. [Ex. XIV, Examples XIV. 1. Find the centre of gravity of the area enclosed between the parabola y = ^mx and the double ordinate corresponding to the abscissa a. 5 2. Find the centre of gravity of the area between the semi-cubical parabola ay^ = x^ and the double ordinate which corresponds to the abscissa a. ' - _ 5^ 3. Find the ordinate of the centre of gravity of the area between the axis of ~v and the sinusoid j = sin x, the limits being .r = o and 4. Find the coordinates of the centre of gravity of the area be- tween the axes and the parabola /\i_ ,7; + ^1) = ■• — a ■, - b A- = — , and v = - . 5 ' 5 5. Find the centre of gravity of the arja between the cissoid y^ (a — x) = x^ and its asymptote. Solution : — Denoting the statical moment by M and the area by A^ -, f" A^ dx I. , ,,n« r« 3. . w , M = = — 2JCS ia — xy- + 1^ •v=' {a — xP dx f" x^ dx I., ,,n a + 5 x-i {a — x) : — 2A-» ya — .vj- = 5'^ • ^^ - 5-1^ ; \M — ^ A, hence -=?• 213 § XIV.] EXAMPLES. 6. Find the centre of gravity of the area between the parabola v^ = 4ax and the straight line 7 = mx. — 8a , - 2« -v = — — and J' = — . Sm m 7. Find the centre of gravity of the segment of an ellipse cut off by a quadrantal chord. — 2a . - 2 b -V = - • and y = - • . I 7t — 2 ■' I 7t— 2 8. Given the cycloid, y — a{i — cos '/•)> X = a {ip — sin rp) , find the distance of its centre of gravity from the base. > = f- 9. Find the centre of gravity of the area enclosed between the positive directions of the coordinate axes and the four-cusped hypo- cycloid .r^ + J^'^^ = Or' Put X = a cos' b, and y = a sin^ 0. 2^(ia x=y = -^ . 315^ 10. Find the centre of gravity of the area enclosed by the cardioid r = \y\oc Use the method of Art. 183. ^ — • 3 a 214 MECHANICAL APPLICATIOXS. [Ex. XIV. 12. Find the centre of gravity of the segment of a circle, the angle subtended being 2« and the radius of the circle a. Solution : — 2 X = - J ^ cola _ 2« Sin a _ Chord Area 3 Area ~ 12 Area" 13. Find the centre of gravity of a circular ring, the radii being a and <7„ and the angle subtended 2 a. - _ 2 a^ — a^ sin ce ~ 3 (f — a^ a 14. Find the centre of gravity of a circular arc, whose length is 2s. Solution : — We have in this case, taking the origin at the centre and the axis of X bisecting the arc. - f^ X — •' -' ~ I: ds Is Put X — a cos 0, then ds = a dO, and denoting by a the angle subtended by s, we have a I cos d^j asm a c 2c being the chord. § XIV.] EXAMPLES, 215 15. Find the coordinates of the centre of gravity of arc of the semi- cycloid whose equations, referred to the vertex, are X =■ a{\ — cos ip), and y =^ a (ip + sin tp). X = — , and y = (tt — - j a. 16. Find the centre of gravity of the arc between two successive cusps of the four-cusped hypocycloid 2. i 2 _ -_ 2« ^~ "J' 17. Find the position of the centre of gravity of the arc of the semi- cardioid r = a {i — cos ^). - 4a . - Aa X = , and y = — . 5 5 18. A semi-ellipsoid is formed by the revolution of a semi-ellipse about its major axis ; find the distance of the centre of gravity of the solid from the centre of the ellipse. X = ^ . 19. Find the centre of gravity of a frustum of a paraboloid of revolution having a single base, A denoting the height of the frustum. - 2/i 20. A paraboloid and a cone have a common base and vertices at the same point ; find the centre of gravity of the solid enclosed between them. The centre of gravity is the middle point of the axis. 2l6 MECHANICAL APPLICATIONS. [Ex. XIV. 21. Find the centre of gravity of a hyperboloid whose height is K the generating curve being y = m {2 ax + -x"). - _ // 8a + zh ~ 4 3^ + '^ " 22. Find the centre of gravity of the solid formed by the revolution of the sector of a circle about one of its extreme radii. The height of the cone being denoted by //, and the radius of the circle by a, we have ^ = |(a+/0. 23. Find the centre of gravity of the solid formed by the revolution about the axis of x of the curve cry = ax^ — a-', between the limits o and a. 8 • 24. A solid is formed by revolving about its axis the cardioid r = a {i — cosO) ; find the distance of the cusp from the centre of gravity. — 16a '^~ 15 ■ 25. Determine the position of the centre of gravity of the volume included between the surfaces generated by revolving about the axis of X the two parabolas y = mx, and y'^ = m' {a — x). - a m + 2m X = 3 m + m § XIV.] EXAMPLES. 217 26. Find the centre of gravity of a rifle bullet consisting of a cylin- der two calibers in length, and a paraboloid one and a half calibers in length having a common base, the opposite end of the cylinder con- taining a conical cavity one caliber in depth with a base equal in size to that of the cylinder. The distance of the centre of gravity from the base of the bullet is \%\ calibers. 27. A solid formed by the revolution of a circular segment about its chord is cut in halves by a plane perpendicular to the chord ; determine the centre of gravity of one of the halves. This solid is called an ogival. Denoting by 2a the angle subtended by the chord, and by a the radius of the circle, the distance of the centre of gravity from the base is — _ d 44 sin- a 4- sin^ 2(t 4- 32 (cos 201 — cos a) 16 sin a (2 -f cos" a) — yx cos a 28. Find the centre of gravity of the surface of the paraboloid formed by the revolution about the axis of x of the parabola a denoting the height of the paraboloid. -. _ I (3^ — 2111) (a 4- my-i 4- 2/«8 5 {a -h my — m-i 29. Find the centre of gravity of the surface generated by the revo- lution of a semi-cycloid about its axis, the equations of the curve being .V = a (i — cos ^), and _y = a (^' 4- sin ^•). 2a Y^TT — % X = — • -^ 15 3^- 4 ' 2l8 MECHANICAL APPLICATIONS. [Ex. XIV. 30. Find the centre of gravity of the surface generated by the revo- lution about its axis of one of the loops of the lemniscata r' zn d' cos 2O. — 2+1^2 X = ; a. 31. A cardioid revolves about its axis ; find the centre of gravity of the surface generated, the equation of the cardioid being r = a {1 — cosO). x-= ^ — , 6; 32. A ring is generated by the revolution of a circle about an axis in its own plane ; c being the distance of the centre of the circle from the axis, and a the radius, determine the volume and surface generated. F= 21V cd\ and S— ^Tt-ca. ^^. A triangle revolves about an axis in its plane ; a^, a^, and a^, denoting the distances of its vertices from the axis, determine the vol- ume generated. 27rA , . V = (^1 + a, + a^). 3 34. Find the volume of a frustum of a cone, the radii of the bases being Ui and a-j, and the height h. •3 35. Find the volume and surface generated by the revolution of a cycloid about its base. y= ^Tta, and o = . XV.] MOMENTS OF INERTIA. 2ig XV. Afomen/s of Inertia. 188. When a body rotates about a fixed axis, the velocity of a particle at a distance r from the axis is dco in which go is the angle of rotation. The force which acting for a unit of time would produce this motion in a mass jh is measured by the momentum doo mr —— . dt The moment of this force about the axis is therefore o doo dt The sum of these moments for all the parts of a rigid system is doj „ , —r -^ A mr' , dt ' since the angular velocity, —p, is constant. In the case of a dt continuous body this expression becomes dco f , , -— xr dm. dt J in which dm is the differential of the mass. The factor r^ djn, 220 MECHANICAL APPLICATIONS. [Art. 1 88. which depends upon the shape of the body, is called its mo- inent of inertia, and is denoted by /. 189. When the body is homogeneous, dm is to be taken equal to the differential of the line, area, or volume, as the case may be. For example, in finding the moment of inertia of a straight line whose length is 2a, about an axis bisecting it at right angles, we let x denote the distance of any point from the axis; then dm = dx, hence we have I=[' ,^dx = '-^=^-''l )-a ^ 12 Again, in finding the moment of inertia of the semi-circle in figure 25, about the axis oi y, let d}n= 2ydx; then, since every point of the generating line is at the distance x from the axis, the moment of inertia is 7=2 yx^ dx = 2 ^/{c? — x^) x^ dx . Jo Jo Putting X = a sin B, we have 1 / = 2a^ \ cos2 B sin2 d dS = ~ . Jo «J T/ie /Radius of Gyration. 190. If the whole mass of the body were situated at the distance k from the axis, its moment of inertia would be Ic^m. Now, if k is so determined that tJiis vwmcnt shall be equal to the actual moment of inertia of the body, the value of k is tJie radius of gyratiofi of the body with reference to the given axis. Hence ^ _ Moment of inertia Mass * § XV.] THE RADIUS OF GYRATION. 221 Thus, for the radius of gyration of the line 2a, whose moment of inertia is found in the preceding article, we have /^ =- , or k= — \ 3 V3 and for the radius of gyration of the semi-circle, whose area is \7lC^, j^ a^ y a k- — ~ , or k= ~. 4 • 2 It is evident that this expression is also the radius of gyra- tion of the whole circle about a diameter, for the moment of inertia of the circle is evidently double that of the semi-circle, and its area is also double that of the semi-circle. 191. It is sometimes convenient to use modes of generating the area or volume, other than those involving rectangular coordinates. For example, let it be required to find the radius of gyration of a circle whose radius is a, about an axis passing through its centre and perpendicular to its plane. This circle may be generated by the circumference of a variable circle whose radius is r, while r passes from o to a. The differential of the area is then 27ir dr, and the moment is I = 27t\ f^ dr = — . Jo 2 Dividing by the area of the circle, we have a^ 2 192. Again, to find the radius of gyration of a sphere whose radius is a about a diameter. In order that all points of the elements shall be at the same distance from the axis 222 MECHANICAL APPLICATIONS. [Art. I92. we regard the sphere as generated by the surface of a cyHnder whose radius is x, and whose altitude is 2y. The surface of this cylinder is therefore 4^,17. The differential of the volume is /^Ttxy dx, and the moment of inertia is x^y dx = /^7i\ V{a- — x) x^dx. Putting X = asin 6, IT I = 47TCV- \\m^ d cos^ e dd J o Dividing by - — , the volume of the sphere, we have 5 Radii of Gyration about Parallel Axes. 193. The moment of inertia of a body about any axis exceeds its moment of inertia about a parallel axis passing tJirougJi the centre of gravity ^ by the product of the mass and the square of the distance between the axes^ Let h be the distance between the axes. Pass a plane through the element dm perpendicular to the axes, and let r and ri be the distances of the element from the axes. Then, r, rj, and h form a triangle ; let 6 be the angle at the axis passing through the centre of gravity, then r^ = r'^ + U^ — 2rji cos B (i) § XV.] RADII OF GYRATION ABOUT PARALLEL AXES. 223 The moment of inertia is therefore I r^ihn — rl dm + H^'m — 2h 7\ cos (^ dm . . . (2 ) Now ri and 6 are the polar coordinates of dm, in the plane which is passed through the element; hence the last integral in equation (2) is equivalent to — 2h X dm. But X dm is the statical moment of the body about the axis passing through the centre of gravity. Nov.' from the defini- tion of the centre of gravity, this moment is zero ; hence, equation (2) reduces to 'T dm = r^ dm + Jrm . . . . . . (3 Introducing the radii of gyration, we have also }^^k~-^h' (4) I94-. As an application of this result, we shall now find the moment of inertia of a cone whose height is //, and the radius of whose base is a^ about an axis passing through its vertex perpendicular to its geometrical axis. Taking the origin at the vertex of the cone, the axis of x coincident with the geo- metrical axis, and a circle perpendicular to this axis as the generating element, we have for the area of this element ;r/^ and for its radius of gyration about a diameter parallel to the given axis, '— . 224 MECHANICAL APPLICATIONS. [Art. I94- The distance between these axes being x, the proposition proved in the preceding article gives an expression for the radius of gyration of the element about the given axis; viz., x^ +— . Replacing r^, in the general expression for / (Art. 4 188), by this expression, and substituting iox dm the differen- tial Tty^ dx, we have {f^^-^fdx, (IX in which j' = ^- . Therefore n and since V — , 3 To find the square of the radius of gyration about a parallel axis through the centre of gravity, we have To find the moment of inertia of a right cone about its geometrical axis wc employ the same generating element as before ; but in this case the square of the radius of gyration is — . Hence 2 7r<7-* V' 2 4 ^=-l\^''-^-T,A/'"' § XV.] RADII OF GYRATION ABOUT PARALLEL AXES. 22$ therefore /= , whence J^ = ^, 10 lO Polai' Moments of Inertia. 195. In the case of a plane area, when the axis of rotation passes through the origin, we have r^ = ;i-2 -[- y^, hence r^ dm ~ {x^ + y-) dtii, therefore /= \.-^ dm + = ,r^ dm + j'^ dm that is, the sum of the moments of inertia of a plane area about tzuo axes in its ozun plane at right angles to each other is eqiial to the moment of inertia about an axis through the origin perpendicu- lar to the plane. I in the above equation is called the polar moment of inertia. In the case of the circle, since the moment is the same about every diameter, the polar moment is twice the moment about a diameter ; that is, denoting the former by Ip and the latter by /,„ we have See Art. 191. T T ^^ Examples XV. I. Find the radius of gyration of a circular arc {2s) about a radius passing through its vertex. 226 MECHANICAL APPLICATIONS. [Ex. XV. Solution : — Taking the origin at the centre, and the axis of x bisecting the arc, and denoting by 2ix the angle subtended by 2s, we have mk ' = [' / (fs = a' [* sin' do. ,„ a f sin 20! m = 2aa .'. ^ = - I i — 2 \ 2a 2. Find the radius of gyration of the same arc about the axis of y, and thence about a perpendicular axis through the centre of the circle. ^ = ^• 3. Find the radius of gyration of the same arc about an axis through its vertex perpendicular to the plane of the circle. See Ex. XIV., 14, and denote by c the subtending chord. k'' = 2a'(i-~ 4. Find the moment of inertia of the chord of a circular arc, in terms of the diameter parallel to it, and its angular distance from this diameter. See Arts. 189 and 193. /=— (3 cos a- — cos3n'). 5. Find the radius of gyration of an ellipse about an axis through its centre perpendicular to its plane. Find the radius of gyration about the major axis and about the minor axis, and apply Art. 195. 6. Find the radius of gyration of an isosceles triangle about a per- pendicular let fall from its vertex upon the base {2b), k' = '- 6' § XV.] EXAMPLES. 227 7. Find the radius of gyration about the axis of the curve, of the area enclosed by the two loops of the lemniscata r = (2' cos 29. ^'=^(3^-8). 8. Find the radius of gyration of a right triangle, whose sides are a and b. about an axis through its centre of gravity perpendicular to its plant' 9. Find the radius of gyration of a portion of a parabola bounded . by a double ordinate perpendicular to the axis, about a perpendicular to its plane passing through its vertex. 10. Find the radius of gyration of a cylinder about a perpendicular that bisects its geometrical axis, 2/ being the length of the cylinder, and a the radius of its base. ,0 ci' I' 4 3 11. Find the radius of gyration of a concentric spherical shell about a tangent to the external sphere, the radii being a and b. 7^;" — 5«V —2b'' k' = S{a^-n 12. Find the radius of gyration of a paraboloid of revolution about its axis, in terms of the radius {b) of the base. ,. b' 13. Find the moment of inertia of an ellipsoid about one of its principal axes. 15 228 MECHANICAL APPLICATIONS. [Ex. XV, 14. Find the radius of gyration of a symmetrical double convex lens about its axis, a being the radius of the circular intersection of tne two surfaces, and b the semi-axis. , _ // + ^d'b^ + \oa* \o{b' + la-) 15. Find the radius of gyration of the same lens about a diameter to the circle in which the spherical surfaces intersect. ^ _ \oa' + x^d'b^ 4- it* THE END. UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. -^94? JUL 'L OCT 8 «4J NOV ^ i 1947 jUN 1 5 195C |iB>ttiill^ ecri4 w^ ^^'V 13/95, iOV 1 9 1^5! APR 2 1 1952 |\U6 2 8 1W3 JUN 4 1954 Form L9-25m-8,'46(9852)444 JAN JUN t 9 1 n JUL 2 7 1955 SEP 1 4 195tf ^m wMm ' ii n ;■■:' iV a :■■■■ , »