MENSURATION whence, by substitution, the volume of prolate frustum - Jirfc^ + ftJ). Similarly we can show that the volume of the oblate frustum where a 1 = I M . These formulae play an important part in the gauging of casks. E. Paraboloid. 101. Surface of a Paraboloid. Let the equation to the para bola be y- = 4ax, and let the coordinates of P (fig. 21, p. 19) be x lt 2/j, then the surface of the paraboloid generated by the revolution of AM about AP , f* 1 A V rf yVj /-/"*> / = 27r/ ,/ f.1 1 + 1 -j }dx=iir/ii/ Ux jo J V dxj v yo v 102. Volume of a Paraboloid. With the same notation we have volume = IT/ l y-dx= lira/ xdx => ITT x 4rta:, x x, = iry x x l ; Jo Jo or the volume of a paraboloid generated by the revolution of a part of a parabola between the vertex and any point is equal to half the volume of the circumscribing cylinder. 103. If the coordinates of Q be x.,, y v then the volume of the frustum PP Q Q = i"- { 2/2*2 - V*x } = Ina&l - x ) = $ir( yl + y J )h , where 7t = MN ; hence the volume of the frustum of a paraboloid is equal to half the sum of the areas of its ends multiplied by its height. F. Ellipsoid. 104. Volume of an Ellipsoid. The equation to the ellipsoid being y? t/ 2 z 2 __ i_ y_ __ i __ _ i rt - + ^ + c"- 1 the equation to the elliptic section at the distance z from the origin is Now if we draw an indefinite number of parallel planes per pendicular to the axis of z, each slice will be an infinitely thin cylindrical plate, and accordingly the whole volume of the ellipsoid =fKdz, where A is the area of the elliptic section. But A = aZl--V 51, therefore volume = irab/ [ 1 - ^ }dz = %-irabc . J-c c / The sphere being an ellipsoid whose axes are all equal, we obtain as before volume of sphere = $ira? = %irr 3 . G. Hyperboloid. 105. Volume of an Hyperboloid. The hyperboloid is generated by the revolution of the hyperbolic segment ANP about AN (fig. 24, p. 20). If the coordinates of P be x : , y v then volume of hyperboloid = it I y-dx = TT-^T / ^ (a: 2 - a*)dx
- Wo, , , N
( ] whence where h=A.T Again, since x lt y-^ is on the curve, we have 2 </! - b"( b- - 2 & 2 , which gives ^ = ^~c L r^ a" " i r i, T i i volume oi hyperboloid 3 + h H. Solids to ivldch the " Prismoidal Formula" applies. 106. It was shown in 72 that the volume of any polyhedron bounded by two parallel planes and by plane rectilinear figures where A 1? A 3 , and A 2 denote respectively the areas of the two ends and of the middle section. We now proceed to show that the same formula determines the volumes of all solids bounded by two parallel planes, provided the area of any section parallel to these planes can be expressed as a rational integral algebraic function of the third degree in x, where x is the distance of the section from either plane. Let 0(a:) = A + Bz -f Cx- + Dx* + .... + Kx n denote the area of the section in question. Now the solid between the sections 0(0) and 0(4) is equal to the solid between the sections 0(0) and 0(2) plus the solid between the sections 0(2) and 0(4). Hence if the prismoidal formula is to hold in this case, we have 40(2) + 0(4)} 40(1) + 0(2)} where h is the distance between the sections 0(0) and <f>( Hence we have 0(0) - 40(1) + 60(2) - 40(3) + 0(4) - . Now 0(0) = A -40(1)= -4A-4B-4C-4D-4E- .... - 4K -40(3)= -4A-12B-36C-108D-324E- ... -4 3"K +4K. Therefore .... +TK. Hence E = F= . . . . K = 0, and therefore 0(x) must be a func tion of the third degree in order that the prismoidal formula may apply. 107. If we take <(>(x) = A + B.C + Cx 2 + Dx 3 , there will be as many possible varieties as there are combinations of four things, one, two, three, and four together, i.e., 2 4 -l = 15 varieties. Corresponding to each of these there will be at least one solid the area of a section of which at a distance x from one of the parallel planes is <f>(x) = A + Bx + Cx 2 + Dx 3 , and at least one solid of revolution generated by the curve whose equation is of the form iry- = <j>(x) = A + Bx + Cx 2 + Da; 8 . As space prevents us discussing all the cases that may arise, we content ourselves by giving three examples as illustrations. (a) Volume of an ellipsoid. Here <p(x) = Bx + Cx 2 . Let 2a, 26, and 2c be the axes of which 2a is the greatest, then /i = 2a, A 1 = 1 , A s = 0, and A 2 = ir6c ; therefore volume = &h( Aj + 4A 2 + A 3 ) = $a(4ir&c) = $irabc , which agrees with the result in 104. (0) Volume of a sphere. Here w?/ 2 = 0(z) = Bx + Car 2 . Let r be the radius of the sphere, then h = 2r, Aj = 0, A 3 = 0, and Aj-TiT 2 , hence, as before ( 91), volume of sphere Of
- + 4A 2 + A 3 ) = (4nr-)
(7) Volume of a right circular cone. Here vy" 2 = <f>(x) = Ca; 2 . Let r = radius of base and h the altitude, then A^O, A 3 = irr 2 , and A 2 = ir(Jr) 2 ; hence volume of cone = JA{A 1 + 4A 2 +A 3 } =i&{jrr 2 + wr 2 } = ^irr 2 In a similar manner we can determine the volumes of a cylinder, a prolate spheroid, an oblate spheroid, &c. 108. In general, if in any solid we have A 2 = A 3 = 0(A) = A + BA + CA 2 + DA 3 , volume of solid = J/t(Aj where A, B, C, and D are known constants, then, if h be the length of the solid, and therefore I. Solids of Revolution in General. 109. Volume of any Solid of Revolution. Let PjPj . . . . P* (fig. 34) be the generating curve, and Aj .... AM the axis of revolution. Divide the curve into portions in the points P 2 , P 3 , &c., and draw the chords and tangents of the small arcs lP z , P 2 P 3 , &c., then it is evident that the solid generated by the curve is greater than the sum of the conical frusta traced out by the chords and less than the sum of the conical frusta traced out by the tangents. Hence, by increasing the number of chords, namely, by increasing the points of division of the curve, we can make tho difference between these sums as small as we please, and therefore by this method we can approximate as closely as we please to the volume of the solid generated. Assuming that the points P a , P 2 , P 3 are so near each other that the solid generated differs little from the frustum of a cone, and using the same notation as in 63, we have volume generated by similarly the volume generated by PsP^- **(! +**;+!);
whence the volume generated by the whole curve P,P 2 . . . . P n