A Course in Mathematical Analysis Volume 1

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 Excerpt: ...is taken correspond according to the convention made above. IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS 137. Volumes. Let us consider, as above, a region of space bounded by the xy plane, a surface S above that plane, and a cylinder whose generators are parallel to the z axis. We shall suppose that the section of the cylinder by the plane z = 0 is a contour similar to that drawn in Fig. 25, composed of two parallels to the y axis and two curvilinear arcs A PB and A 'QB'. If z = f(x, y) is the equation of the surface S, the volume in question is given, by § 124, by the integral 'dx / f(x,y)dy. Now the integral frf(x, y)dy represents the area A of a section of this volume by a plane parallel to the yz plane. Hence the preceding formula may be written in the form (39) V==fa kdx' The volume of a solid bounded in any way whatever is equal to the algebraic sum of several volumes bounded as above. For instance, to find the volume of a solid bounded by a convex closed surface we should circumscribe the solid by a cylinder whose generators are parallel to the z axis and then find the difference between two volumes like the preceding. Hence the formula (39) holds for any volume which lies between two parallel planes x = a and x = b (a b) and which is bounded by any surface whatever, where A denotes the area of a section made by a plane parallel to the two given planes. Let us suppose the interval (a, b) subdivided by the points a, xlf xt, xn_x, b, and let A0, Au » A,» be the areas of the sections made by the planes x = a, x = xlt respectively. Then the definite integral fkdx is the limit of the sum A0(i-a)+ Ai(x2-aH h A.-.x,---x)--. The geometrical meaningof this result is apparent. For A_i(--x,_i), for instance, represents the volume of a right cylind...