An elementary course of infinitesimal calculus . x, say /(^)=^-f5| + 0| (1). Integrating this between the limits 0 and h, we find /: f(a;)dx = h{A + ^B + iG) (2). 107] GEOMETRICAL APPLICATIONS. 261 Now A is evidently the area of the section a; = 0; anddenoting the areas of the sections x = ^h, and x = h, respec-tively, by A and A,Yie have A+i,B-\r\G=A, A + B + G = A (3), whence pf(x) dic = lh(A+4!A+ A) (4). Jo This gives the volume included between two parallelsections, in terms of the areas of these sections and of thesection half-way between them. The formula (1) is obviously applicable to t
An elementary course of infinitesimal calculus . x, say /(^)=^-f5| + 0| (1). Integrating this between the limits 0 and h, we find /: f(a;)dx = h{A + ^B + iG) (2). 107] GEOMETRICAL APPLICATIONS. 261 Now A is evidently the area of the section a; = 0; anddenoting the areas of the sections x = ^h, and x = h, respec-tively, by A and A,Yie have A+i,B-\r\G=A, A + B + G = A (3), whence pf(x) dic = lh(A+4!A+ A) (4). Jo This gives the volume included between two parallelsections, in terms of the areas of these sections and of thesection half-way between them. The formula (1) is obviously applicable to the case of a cone,pyramid, or sphere, and also to the case of a paraboloid,ellipsoid, or hyperboloid, provided the bounding sections beperpendicular to a principal axis. The student who is familiarwith the theory of surfaces of the second degree will easily convincehimself, moreover, that the latter condition is not essential. Another case coming under the present rule is that of asolid bounded by two parallel plane polygonal faces and by plane. Fig. 66. lateral faces which are triangles or trapeziums. We may eveninclude the case where some or all of the lateral faces are curvedsurfaces (hyperbolic paraboloids) generated by straight hnesmoving parallel to the planes of the polygons, and each inter-secting two straight lines each of which joins a vertex of onepolygon to a vertex of the other (see Fig. 67). And since thenumber of sides in each polygon may be increased indefinitely,the rule will also apply to a solid bounded by any two plane 262 INFINITESIMAL CALCULnS. [CH. VII parallel faces and by a curved surface generated by a straightline which meets always the perimeters of those faces.
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