Elements of the integral calculus; with a key to the solution of differential equatons, and A short table of integrals - Softcover
Synopsis
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. 1889 Excerpt: ...+ df), and greater than a prism having the same base and the altitude 2 irr sin f; and these prisms differ by an amount which is infinitesimal of higher order than the second. We shall have then F= 2 irffr sin fdrdl, 1 the limits being so taken as to bring in the whole of the generating area. For example; let us find the volume generated by the revolution of a cardioide about its axis.?= 2a(l--cos is the equation of the cardioide; Example. A right cone has its vertex on the surface of a sphere, and its axis coincident with the diameter of the sphere passing through that point; find the volume common to the cone and the sphere. Volume of any Solid. Triple Integration. 153. If we suppose our solid divided into parallelopipeds by planes parallel to the three coordinate planes, the elementary parallelopiped at any point (x,y,z) within the solid will have for its volume AxAyAz, or, if we regard x, y, and z as independent, dxdydz; and the whole volume V=j'j'j'dxdydz, 1 the limits being so chosen as to embrace the whole solid. The integrations are independent, and may be performed in any order if the limits are suitably chosen. As it is important to have a perfectly clear conception of the geometrical interpretation of each step in the process of finding a volume by a triple integration, we will consider one case in detail. Let the integrations be performed in the order indicated by the formula r r f If the limits are correctly chosen, our first integration gives us the volume of a prism one of whose lateral edges passes through any chosen point P,(x,y,z) within the solid, is parallel to the axis of Z, and reaches directly across the solid from surface to surface, while the base of the prism is the rectangle dydx; our second integration gives the volume of a right...
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