CHAPTER X. CUBATURE OF SOLIDS. Section 1.—Cubature of Solids of Revolution ; and of similarly generated Solids. 251.] Another important application of definite multiple integration is that whereby we are able to determine the volume, or the quantity of space, contained within given bounding surfaces, and thereby to compare it with the volume of a cube; whence arises the name Cubature of Solids. The process will generally involve triple integration; and I shall first investigate the most simple case, that viz. of the volume contained within a surface of revolution, and between two given planes which are perpendicular to the axis of revolution. Let OPQBA, see fig. 17, be the plane surface by the revolution of which about the line on, which is the x-axis, the volume is generated; and let the equation to OPQB, which is its bounding curve, be y = f(x). Let E be the point (x, y) in the surface on which abuts the infinitesimal surface-element, the area of which is dy dx ; now imagine the surface to have revolved about the line ox through an angle ; then as is increased by do, the surface element at E will generate a volume, which is approximately a parallelepipedon, of which the base is dy dx and the altitude is ydd ; and consequently the volume of it = y do dy dx ; which is a triple differential, the definite integral of which with the proper limits will give the volume of the solid. To consider the effects of the several integrations. The p-integration with the limits 27 and 0, will give the volume of the ring which is generated by the complete revolution about ox of the surface-element whose area is dy dx. The subsequent y-integration with the limits of the ordinate of the curve and of zero, will give the sum of all these rings which are in a plane slice perpendicular to ox; and this sum is evidently the circular plate of thickness dx which is generated by the revolution of the slice MQ of the generating surface : and the final x-integration will give the sum of all these circular plates, which is the required volume. Thus, if v = the required volume, Axon 71(x) 821 = 2*1**/ y dy dx == )* {f(x)}” dx. This process resolves the problem of the cubature of a solid of revolution into its most simple elements, and is applicable when the generating area does not revolve through a whole circumference, and in that case the limits of the p-integration will not be 27 and 0, but on and do say, which will be given by the conditions of the problem. The process is also applicable when the limits of the y-integration are both ordinates to certain given curves ; in which case the solid of revolution is hollow. Examples of this latter case will occur in the sequel. The final integral given in (3) may however be found by the following process without the intervention of the first two integrations. Let, as before, y = f(x) be the equation to the plane curve bounding the area, by the revolution of which the solid is generated; and let the axis of x be that of revolution; see fig. 33 ; then, as the elemental area PQNM revolves about ox, it generates a circular slice whose thickness is MN=dx, and of whose circular faces one has a radius MP = y, and the other has a radius NQ = y +dy; therefore the volume of the slice is intermediate to TYP dx and T(y+dy) dx ; whence, neglecting infinitesimals of the higher orders, as is necessary, the volume of the elemental slice is equal to ty? dx ; and therefore, if X, and X, are the distances from the origin of the extreme faces of the solid thus formed, the volume = * / **y dx (4) 252.] The following are examples in illustration of the preceding formula. Ex. 1. To find the content of a right circular cone, whose altitude is a, and the radius of whose circular base is b. The equation to the generating line is y = x; therefore Thus the volume of the cone is equal to one-third of that of the cylinder of equal altitude and equal base. Ex. 2. To find the content of a paraboloid of revolution whose altitude is a, and the radius of whose base is b. The equation to the generating curve is ay = 62x; and 62 pa 7b2a the volume of the paraboloid = 1.63*/*dx = **. Thus the volume of the paraboloid is one-half of that of the circular cylinder on the same base and of the same altitude. Ex. 3. To find the volume generated by a circular area of radius = a, revolving about the diameter, the abscissæ to the extremities of which are X, and X. Here, since x + y2 = a’, the volume = * /*(a? – x2) dx = n {d? (n–co) – (0,3–2*)}. Hence the volume of a spherical segment = */*ca? —xo) dx ne and the volume of the sphere = 2- 4*2* —) dx = tao. Ex. 4. The volume of a prolate spheroid = Tab. Ex. 5. Determine the volume formed by the revolution about its base of the cycloid, whose equations are x = a (0-sin 8), y=a(l- cos ). le Ex. 6. The volume generated by the revolution of the cycloid about its axis is Ex. 7. The volume of the solid generated by the revolution of y = a log x, about the axis of x, with the limits x and 0, = 7a$ x {(log x)2 — 2 log x+2}. Ex. 8. The volume of the solid generated by the revolution of y = q* about the axis of X, with the limits x and -00, = + (log a)-122x. Ex. 9. If the equation to the bounding curve of the revolving area is xy = k', the volume of the solid = 7 k2 (4.-yn); and if Yx= 0, the volume = a ko yo. Ex. 10. Find the curve which by its revolution about the L-axis generates a solid, the volume of which varies as the nth power of (1) the extreme ordinate ; (2) the extreme abscissa. Ex. 11. The volume of the solid generated by the revolution πα3 of the equitangential curve round its x-axis = " 253.] If the plane area revolves about the axis of y, and thereby generates a solid, then, as is manifest from fig. 34, the volume of the solid = * / xạ dy ; (6) të dy being expressed in terms of a single variable by means of the equation to the bounding curve, and the limits being assigned by the conditions of the problem. Ex. 1. Determine the volume of an oblate spheroid. Since Ex. 3. The equation to the Witch of Agnesi being y2 = 4 a2 2a=, the volume of the solid formed by its revolution about the asymptote is 4 72 a. Ex. 4. The volume of the solid generated by the revolution of the right-angled triangle whose sides are a, b, c about a line passing through a and parallel to bc = "790. Ex. 5. The volume of the solid generated by the revolution of the companion to the cycloid about its base = 3 ma'. 254.] If the plane surface by the revolution of which the solid is formed, is referred to polar coordinates, and the prime radius is the axis of revolution, then, see fig. 26, the area of the infinitesimal area-element abutting at E is rdr do, and the perpendicular distance of E from the prime radius is rsin 0; so that if the plane surface revolves about ox through an angle do, it generates an approximate parallepipedon of which the base is r dr do, and the altitude is r sin 0 do, and consequently the volume of it = 7% sin 0 do dr do; and the triple definite integral of this differential will give the volume of the generated solid. Now the 0-integration with the limits 27 and 0 will give the volume of the ring generated by the complete revolution of the surfaceelement whose area = r dr do; the r-integration, with the limits of the radius vector to the curve and of zero, will give the sum of all these rings, which is evidently a conical shell, the thickness of which varies directly as the distance from the vertex; and the final 6-integration, with the limits assigned by the problem, will give the whole required volume. If these final limits are 0, and 0, the solid will be full; but if the inferior limit is a finite angle, say 0o, the solid will be hollow. Thus if v is the required volume, and r=f(0) is the equation to the bounding curve, for pr 127 @ vodo |