A Course in Mathematical Analysis Volume 1 (Paperback)

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 edition. Excerpt: ...and contain all the vertices.--Trans. a ruled surface, and the area of the section made by a plane parallel to the plane x = 0 is given, by 94, by the integral where a', 6', c', d' denote the derivatives of a, 6, c, d with respect to t. These derivatives may even be discontinuous for a finite number of values between to and T, which will be the case when the lateral boundary consists of portions of several ruled surfaces. The expression for A may be written in the form A = x2 f ab'dt + x f (aq' + pbr)dt 4-f pq'dt, where the integrals on the right are evidently independent of x. Hence the formula (40) holds for the volume of the given solid. It is worthy of notice that the same formula also gives the volumes of most of the solids of elementary geometry. 139. Viviani's problem. Let C be a circle described with a radius OA ( = K) of a given sphere as diameter, and let us try to find the volume of the portion of the sphere inside a circular cylinder whose right section is the circle C. Taking the origin at the center of the sphere, one fourth the required volume is given by the double integral extended over a semicircle described on OA as diameter. Passing to polar coordinates p and w, the angle w varies from 0 to Tc/'2, and p from 0 to B cos w. Hence we find Replacing p and q by their values--x/z and--y/z, respectively, and passing to polar coordinates, we find o Jo Vfi7 J- or fi = 4R2 f2(l-sma)da=:4K(--lY Subtracting the area enclosed by the two cylinders fiom the whole area of the sphere, the remainder is 4ri?2-8W--l) = 8R2. 140. Evaluation of particular definite integrals. The theorems established above, in particular the theorem regarding differentiation under the integral sign, sometimes enable us to evaluate certain definite...