. Applied calculus; principles and applications . ~\ = 2 tt Hnear — The volume of the cone of Art. 155 may be graphi-cally represented by the area under the parabola y = -r^x^, and the volume of the sphere by the area under the parabola2/ = TT (a^ — x^). If the first integral curves. An 3^,x3 and = .(a^x-|) be drawn, their ordinates will represent both the areas and thevolumes in the two cases, respectively. 157. Surface and Volume of Any Frustum. — A solidbounded by two parallel planes is, in general, called afrustum. One or both of the truncating planes may inspecial cases, as in
. Applied calculus; principles and applications . ~\ = 2 tt Hnear — The volume of the cone of Art. 155 may be graphi-cally represented by the area under the parabola y = -r^x^, and the volume of the sphere by the area under the parabola2/ = TT (a^ — x^). If the first integral curves. An 3^,x3 and = .(a^x-|) be drawn, their ordinates will represent both the areas and thevolumes in the two cases, respectively. 157. Surface and Volume of Any Frustum. — A solidbounded by two parallel planes is, in general, called afrustum. One or both of the truncating planes may inspecial cases, as in the sphere, touch the frustum in onlyone point and be tangent planes. The method of dividing the soHd into thin slices and takingthe sum of the approximate expressions for the small partsas an approximate expression for the whole, and taking the SURFACE AND VOLUME OF ANY FRUSTUM 271 limit of the sum as an exact expression for the whole, may beapplied to any solid even when the solid is not regular andthe sections not regular plane Let the solid represented in the figure be divided intoslices by planes perpendicular to an axis OX; then, takingAx^x as an approximate expression for the volume of thesHce P — NiMiRi, Ax being the area of the section PNMRat a distance x from plane ZOY, the approximate expressionis x=h X^v = XA.^x, where h{= OA) is the distance between the truncating orbounding planes. The exact expression is V = \imy,Ax^x= I Axdx. (1) A„^n-*-< Jx=0 Ax=0 x=0 When the area of a section is a function of the distance xfrom one of the bounding planes and hence Ax can be ex-pressed in terms of x, the limit may be found by frustum formula for volumes is, therefore, Jr*x = h px=h F{x)dx or V= Ix=0 Jx=hi F{x)dx, (10 272 INTEGRAL CALCULUS where Ax is F(x), some function of x; the one form givingthe whole volume and the other a segment or any partthereof. To get expressions for the area of the surface S, let P bethe curve NPR, then AS = NP
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