INFINITESIMAL C A L U L U S 55 Hon is reversed, vc get two or more double integrals instead of tlie original integral. 176. It is frequently found necessary to transform a double integral jfffavWxdy, referred to rectangular coordinates, to another referred to polar coordinates. In this case, as in 164, we substitute rdrdd instead of dxdy, and the integral becomes ffjr cos 9, ; sin 0)rdrdd The limits in the latter integral are determined from the equa tions which give the limits in the former. For example, to find the volume comprised between the plane of xy, the hyperbolic paraboloid cz = xy, and the right cylinder (x-a)- + (y-l)- = k*. Here V = f/xydxdy , extended to all values of x, y, subject to the condition (x-af + (y-b)-<k. Assuming the origin of polar coordinates at the point a, b, and transforming the equation, we get /ZTT /-. Ti- /-- T / sin6dd = 0,/ cosfldfl O, / sin 0cos0 J0 = 0. /(> ~/0 -M) 177. The triple integral _/#fc, y, z)d X dydz can be transformed in like manner. For, first, take x = p cos <j>, y = p sin 0, and the integral transforms into ffff(? cos </>, p sin (p, z)pdpdzd<j) . Again, assume z = r cos 0, p~r sin 0, and the multiple integral becomes ffff(r sin cos <f>, r sin sin <p, r cos 6)r 2 sin 6 drdQd<j> . "With respect to the limits in the new integral, it may be ob served that, in this and all other cases, the new limits must bo taken in such a manner that the transformed multiple integral shall comprise every element which enters into the original integral, and no more. In particular the volume of any solid is represented by fffi 11 sin 9 drdQd<j> , taken between limits determined by the boundary of the solid. If this expression be integrated with respect to r, we have in which we must substitute for r its value determined by the equation of the bounding surface. For example, let us investigate the volume within the surface (* 2 + y" + ~"} 5 = ( 3 a 2 + s 2 Here we get and, as the equation is symmetrical, we have 4ff , ,., , o. = TT ft + & + C ) i) Again, the expression for the volume of the ellipsoid is represented by the integral 2/2 o sin 2 cos 2 siri 2 sin 2 <j sin 9 d0d<j> p Hence, since the volume of the ellipsoid is sin d0d<j> c, we get > sin 2 sii> 2 cns - fl James500 (talk) 178. The multiple integral admits of a like transformation. For, let x l = p cos </>, x., = p^ sin <p, and it becomes where V x represents the transformed value of V. In like manner, if x 3 = p. 2 cos ^, a. 4 =p 2 sin |>, the transformed integral may be written Again, if p 1 = r cos 9, p. 2 = r sin 6, the integral assumes the form
- sin cos 6
a result which admits of a direct demonstration. where V 3 represents the final form of V. In this case the values of x 1} x. a x 3 , 4 , in terms of the new variables, are x 1 r cos cos <, x 3 =*r sin 9 cos x^ r cos 9 sin <f>, x-~r sin sin Quadrature of Surfaces. 179. It is readily shown that the area of any cylindrical surface, bounded by two planes perpendicular to its axis, is equal to the rectangle under the height of the cylinder and the perimeter of its base ; also that the surface of a truncated right cone is equal to the rectangle under its mean section and the length of the portion of any edge of the cone intercepted between the bounding sections. In the evaluation of the superficial area of a solid of revolution, we proceed, as in 172, by supposing the surface divided by planes perpendicular to the axis of revolution (fig. 11). Then the elementary portion of surface between two indefinitely near planes may be regarded as a portion of the surface of a right cone, generated by the revolu tion of the corresponding ele ment of the curve round the o axis. Hence, denoting the ele ment PQ by ds, and PM by y, Fig. 11. the area generated by PQ in a complete revolution round the axis of x is represented in the limit by 2iryds. Consequently, if S be the surface generated by the curve A15, we have taken between limits corresponding to the points A and B. (1) Thus for the sphere, generated by the revolution of the circle x- + y^a z round the axis of x, we have
- te- &!
y Hence S2ir/Kfa: 2ira(X- ), if X, x be the limits for x. Accordingly, the whole surface is 47ra 2 , i.e., four times the area of one of the great circles of the sphere. Also the surface bounded by any two parallel planes is equal to the corresponding surface cut out of the circumscribed cylinder, whose axis is perpendicular to the bounding planes, (2) If the ellipse revolve round the axis of x, we have 7.4 r o 5 ~ a* y- 1 dx , / where e is the eccentricity of the ellipse. Hence, the whole surface of this ellipsoid is b r - / ab In like manner, if S be the surface generated by the revolution of the ellipse round its axis minor, we get S = 2-n-fxds = ITT^JT, /(b* + tftfy*)dy . Consequently its entire surface is represented by 180. In connexion with surfaces of revolution, the following general propositions, usually called Guldm s theorems, may be here stated. (1) If a plune curve revolve round any external axis situated in its plane, the area of the surface generated in a complete revolution
