4. Transform 0 [JV da dy, where If the limits of y be 0 and x and the limits of x be α -(x2+2xy cos a+y2) dx dy from rectangular to 5. Transform fe-+ polar co-ordinates, and thence shew that if the limits both of x and y be zero and infinity, the value of the integral will be α 2 sin a 6. Prove by transforming the expression from rectangular to polar co-ordinates that the value of the definite integral e-(x2+2x2y2 cos a+y1) dx dy denotes a is equal to √/ F(sin) where F(sin complete elliptic function of the first order, of which 7. Apply the transformation from rectangular to polar coordinates in double integrals to shew that 18 a dx dy = 2π (x2 + y2+a2)a (x2 + y2+a2) $ a+a'' 8. Transform the double integral [[f(x, y) dx dy into one in which rand shall be the independent variables, having given x = r cos 0 + a sin 0, y=r sin 0 + a cos 0. Result. î(r cos 0 + a sin 0, r sin 0+a cos 0) (a sin 20 − r) dė dr. 9. Transform [[e-- de dy into a double integral where r and t are the independent variables where r2 = x2 + y2; and if the limits of x and y and ∞, y X = t and be each 0 10. If x and y are given as functions of r and 0, transform the integral [] da dy dz into another where r, ◊ and z are the variables; and if x = cos 0 and y = r sin 0, find the volume included by the four surfaces whose equations are r = a, z=0, 0 =0, and z = mr cos 0. 11. If ax = yz, By = zx, yz=xy, shew that ||| fla, B, v) đa đảng = 1 SSS T. I. C. X1 'yz zx xy dx dy dz. 2 x y Z [[[[v dxdxdxdx V dx, dx, dx, dx, to r, 0, 4 and when = r sin 0 cos 0, =r sin 0 sin 4, Result. Sv 13. Find the elementary area included between the curves (x, y) = u, ↓ (x, y) = v, and the curves obtained by giving to the parameters u and v indefinitely small increments. Find the area included between a parabola and the tangents at the extremities of the latus rectum by dividing the area by a series of parabolas which touch these tangents and by a series of lines drawn from the intersection of the tangents. 14. Transform the triple integral [ƒƒ.ƒ(x, y, z) dæ dy dz into one in which r, y, z are the independent variables, having given (x, y, z, r) = 0; and change the variables in the above integral from x, y, z to r, 0, 0, having given ¥ (x, y, z, r) = 0, ¥1 (y, z, r, 0) = 0, Y2 (z, r, 0, 0) = 0. in which x, y, z are connected by the x = sin & √√(1 — m2 sin2 0), equation and 4, y = cos e cos p, m2 + n2 = 1. √(1 - m2 sin" 0) √(1-n2 sin' dx 16. Transform the integral [[[da dy dz to r, 6, 4 where x = r sin √√(1 — n2 cos2 ), y = r cos & sin 0, z=r cos 0 √√(cos2 + n2 sin2 p). 22 {(n2 - 1) cos2 - n2 sin2 0 dr de do Result. (1-n2 cos3 e) √(cos 17. Transform the expression [f to rectangular co-ordinates. 3 +n2 +n" sin2 ) sin e do do for a volume - px -qy) dx dy; this should be in Result. (z-p terpreted geometrically. 18. Prove that 19. If (See Arts. 263 and 66; and transform as in Art. 242.) 1 where Vis any function of x,,,,... x, and V' what this function becomes when the variables are changed. 212 CHAPTER XII. DEFINITE INTEGRALS. 252. WHEN the indefinite integral of a function is known, we can immediately obtain the value of the definite integral corresponding to any assigned limits of the variable. Sometimes however we are able by special methods to assign the value of a definite integral when we cannot express the indefinite integral in a finite form; sometimes without actually finding the value of a definite integral we can shew that it possesses important properties. In some cases in which the indefinite integral of a function can be found, the definite integral between certain limits may have a value which is worthy of notice, on account of the simple form in which it may be expressed. We shall in the present chapter give examples of these general statements. 253. Suppose f(x) and F(x) rational algebraical functions of x, and f(x) of lower dimensions than F(x), and suppose the equation F(x) = 0 to have no real roots; it is required to find the value of It will be seen that under the above suppositions, the expression to be integrated never becomes infinite for real values of x. Let a+B(-1) and a-B(-1) represent a pair of the imaginary roots of F(x)=0; then the corresponding quadratic |