A Course in Mathematical Analysis Volume 1

by Douard Goursat

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 ...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, xx, x2, x„_lt b, and let A0, A1;, A,, be the areas of the sections made by the planes x = a, x = xx,, respectively. Then the definite integral JA dx is the limit of the sum A0(a;1-a) + (x2--xx)-h Aas,--x)-. The geometrical meaning of this result is apparent. For A,_! (xt--x), for instance, represents the volume of a right cylinder whose base is the section of the given solid by the plane x = as4_, and whose height is the distance between two consecutive sections. Hence the volume of the given solid is the limit of the sum of such infinitesimal cylinders. This fact is in conformity with the ordinary crude notion of volume. Now the integral Jj'fix, 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 If the value of the area A be known as a function of x, the volume to be evaluated may be found by a single quadrature. As an example let us try to find the volume of a portion of a solid of revolution between two planes perpendicular to the axis of revolution. Let this axis be the x axis and let z = f(x) be the equat...

142 pages, Paperback

First published January 1, 2006