56 INFINITESIMAL CALCULUS equals the product of the length of the generating curve into the path described by its centre of gravity. (2) In like manner, the volume of the solid generated is equal to the product of the generating area into the path described by the centre of gravity of the area. The former of these theorems is easily shown ; for if y l be the distance of the centre of gravity of the curve from the axis of re volution, taken as that of x, we have . . 27T2/JS = 2-irfyds = S , which proves the theorem. Next, if ?/., be the ordinate of the centre of gravity of the area A, we have A?/ a = 2ydA. -=ffydxdy = whence the latter theorem follows. 181. In the general case of the determination of the quadrature of a surface we regard it as the limit of a number of indefinitely small elements, each of which is considered a portion of a plane that is ultimately a tangent plane to the surface. Now let dS denote such an element at any point of the surface, and da- its pro jection on a fixed plane, which makes the angle with the tangent plane at the point, then we have dcr^dS cos 0, or rfS = sec da-. Hence S=_/"sec Gda; taken between proper limits. If now the surface be referred to a rectangular system of coor dinate axes, we may take da*=dxdy also, from an elementary theorem in surfaces,
- , dz dz
sec = J +P + <?> where P^^,* V"~dv Hence we have in which the values of p and q are to be determined from the equa tion of the surface. (1) For example, let it be proposed to find the portion of the sur face of a sphere intercepted by a cone of the second order, whose vertex is on the surface of the sphere, and whose internal axis passes through the centre of the sphere. Let the vertex of the cone be taken as the origin (fig. 12), and the line join ing it to the centre of the sphere as axis of z, then the equation of the sphere may be written x- + y- + 2 = 2az . Hence dz_ _x dz y dx a z dy a z Fig. 12. consequently S = a// ^ - JJ Va 2 -*"-,/* in which the limits are determined from the equation of the bound ing cone. Let the equation of this cone be then, eliminating z, the limiting values of x and y are connected by the equation Next, transform to polar coordinates by making x=r cos 0, y*=r sin 0, and we get rdrdQ taken for all points Within the curve r{(l + A 2 ) cos 2 + (1 + B 2 ) sin 2 0} = Hence, since the curve is symmetrical, we get T~/~ u rdrdQ where 1! /o Jo 2VA 2 cos* Aam f 11 L JO JO (I + A -) cos - + (1 + B 2 ) sin -"0 V(l + A 2 )(l + B-) This result admits of a simple geometrical representation ; for let D, E (lig. 12) be the points in which the edges of the cone lying in the planes y = Q and x = cut the surface of the sphere, and we plainly have rT> ^ a -*U 7" - ( n == ~i 7 * Consequently the area of the intercepted portion of the sphere is equal to that of the ellipse which has CD and CE as its semi-axes. (2) If, instead of the cone, we had taken the paraboloid z = Ax 2 + B?/ 2 , the area of the portion Intercepted on the sphere is given, as in the preceding, by the equation where, from the equation of the bounding curve, we have m 2a( A cos 2 + B sin 2 0) - 1 (A cos 2 + B sin- 0) 2 5 = 4a/ T Jo Hence u = tu,/ A cos 2 + B sure /AB This result admits of a geometrical interpretation similar to that in example (1). Multiple Integrals. 182. The general form of a multiple integral may be represented by the expression J^ dxf dy. . . l~ duf dtf(x, y, . . . u, t) , in which f(x, y, . . . u, t) is supposed continuous for all systems of values of the independent variables x, y, . . . u, t included within the limits. Moreover the limits of each variable must be independent of the following variables, but may depend on the preceding variables. In calculating the integral, the expression f(x, y, . . . u, t)dt is integrated between the limits T and < , regarding x, y, . . . u as constants. Thus we obtain a function of a-, y, . . . u. This func tion is integrated with respect to u between the limits U and u , treating x, y ... as constant. We thus obtain a function of x, y, ... independent of u, t ; and so on for the subsequent integrations. If the limits for each variable be constant, the integrations may be taken in any order, subject to such limitations as those given iii 148 for two variables. In the more general case, when the order of integration is altered it is necessary to determine, from the con ditions of the problem, the new limiting values. This is usually a matter of much difficulty. 183. Continuing from 178, the general problem of the trans formation of a multiple, integral by a change of variables may be stated as follows. Suppose the multiple integral represented by ff. . .y~<f>(x 1 , x.,, . . . x n )dx 1 dx. 2 . . . dx n , and it be proposed to transform it into another, depending on new variables u lt u. 2> ... , which are related with the original variables by a system of n given equations. This transforma tion implies three parts in general: (1) the determination of <f>(x lf x z , . . . x n ) in terms ofu 1} u 2 , . . . u n ; (2) the determination of the new system of limits ; (3) the finding the substitution for The solution of the first two questions is an algebraical problem, of which we have already considered one or two elementary cases. We now address ourselves to the third question, and write the integral in the form In the integration with respect to x n , as stated in 182, a-j, x zt . . . x n -i are regarded as constants. Accordingly, in order to replace x n by u n , it is sufficient to express x n in terms of ?/, x lt dx x. 2 , . . . x n -i, and then to substitute -r^du n for dx n . Again, CtlCn to transform the next integration, relative to d.v,,-, we suppose x n -i expressed in terms of u n -i, u n , x lt x. 2 , . . . x,,.z, and we re place (fa,,-i by n ~ l du,,-. By continuing this process the integral finally becomes of the form
