An Elementary Treatise on the Differential Calculus Founded on the Method of Rates or Fluxions (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1880 Excerpt: ...for the integral involves imaginary quantities; but putting b + - the integral becomes dy Now, if in equation (i) we put for and change the sign of c, we obtain the product of which gives (i + e cos ) (i--e cos ) = I--f.... (7) By means of these relations any expression of the form f J(i + e coso)"' where is a positive integer, may be reduced to an integrable form. For f _ t dB __ i J(i + e cos )" J i + e (i + e cos )'1' hence, by equations (5) and (7), By expanding (i--cos )""1, the last expression is reduced to a series of integrals involving powers of cos; these may be evaluated by the methods given in this section and Section VI, and the results expressed in terms of by means of equation (2) or of equation (7). By adding and subtracting an undetermined constant, the fraction may be written in the form / cos x + q sin x + A (a cos x + b sin x). 49. Find the area of the trochoid (& a) x = atp--6 sin ip y = a--b cos ip. (2a + P) n. 50. Find the area of the loop, and also the area between the curve and the asymptote, in the case of the strophoid whose polar equation is r = a (sec 0 tan 6). Solution: --Using 0 as an auxiliary variable, we have I sin8 61 x = a (1 sine) y = a tan 6, L_ cos 0_ the upper sign corresponding to the infinite branch, and the lower to the loop. Hence, for the half areas we obtain + a' sin 6 afe + a' sin'0 do-a' 1 + and--a8 sin 8 d6 + a f sin" 6 d9 = o' I 1---I. CHAPTER II. Methods Of Integration--Continued. IV. Integration by Change of Independent Variable. 37. If x is the independent variable used in expressing an integral, and y is any function o...