. Differential and integral calculus, an introductory course for colleges and engineering schools. f revolution generated is termed an ellipsoid of revolution,a hyperboloid of revolution, or a paraboloid of revolution, accordingto the species of the conic. The sphere is a particular case of theellipsoid of revolution. When the ellipse is revolved about itsmajor axis, the resulting ellipsoid is sometimes termed a prolatespheroid, when about its minor axis, an oblate spheroid. Whenthe hyperbola is revolved about its transverse axis, the resultinghyperboloid is termed an hyperboloid of two sheets
. Differential and integral calculus, an introductory course for colleges and engineering schools. f revolution generated is termed an ellipsoid of revolution,a hyperboloid of revolution, or a paraboloid of revolution, accordingto the species of the conic. The sphere is a particular case of theellipsoid of revolution. When the ellipse is revolved about itsmajor axis, the resulting ellipsoid is sometimes termed a prolatespheroid, when about its minor axis, an oblate spheroid. Whenthe hyperbola is revolved about its transverse axis, the resultinghyperboloid is termed an hyperboloid of two sheets, when aboutits conjugate axis, an hyperboloid of one sheet. 175. Volumes of Solids of Revolution. Let A B be an arc of the curve y = f(x), and let Vbe the volume of the solid gen-erated by revolving about OX thecurvilinear quadrilateral seek an expression for V. To fix ideas, we suppose thatf(x) is an increasing functionthroughout the interval a to is assumed, of course, that f(x)is single-valued, continuous, andreal throughout this interval. Divide the interval a to 6 into n 253. 254 INTEGRAL CALCULUS §175 subintervals, erect ordinates at the points of division, and completethe interior rectangles as shown in the figure. As the plane re-volves about OX, each of the curvilinear quadrilaterals, such asPQST, generates a solid of revolution whose volume we denoteby Vq, and it is obvious that V is the sum of the volumes Vq what-ever be the value of n. Now let n increase indefinitely by re-peated subdivision of each subinterval of the a>axis, and we haveof course b (a) V = lim V Vq. We shall now show that in the second member of this equationeach Vq may be replaced by Cp, the volume of the cylinder gen-erated by the corresponding interior rectangle PRST. On thebase ST or Ax, construct the exterior rectangle PQST, and denotethe volume of the cylinder which it generates by Cp>. Then itis geometrically evident that CP < VQ and 1< -^ = ir(y + Ay)2Ax, and there
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