A Rudimentary Treatise on the Integral Calculcus. (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. 1852 Excerpt: ...the quadrature, the terms involving different values of any variable may be first collected, and the limit of their sum involves an integral with respect to that variable. Therefore, the variable with respect to which the first integration is performed, is indifferent. Similar reasoning applies to the other integrations. Corollary. / dy ( f f(z, y)dz) 119. The cubature of solids affords a very complete illustration of the foregoing principles. Let xOz, xOy, yOz be three planes perpendicular to each other; and let ABCDabed, be a solid bounded by the curved surface ABCD, by a rectangle ac in the plane xOz, by two planes Ab, Be parallel to the plane yOz, and two planes Ad, Be parallel to the plane xOz. Consider now the base ac of the solid divided into any number of rectangles, represented by dotted lines in the figure, and on these rectangles, as bases, let rectangular parallelopipeds be described, of which the sides cut the upper surface ABCD in the curves shewn in the diagram. If x, y, z be co-ordinates of any point (P) in the curved surface referred to rectangular axes Ox, Oy, Oz, the relation between x, y, z may be expressed by an equation in which z is supposed to be finite and continuous; and pq = x, Oq = y, Vp = z. Let Vp be the altitude of one of the elementary parallelopipeds, 8x and $y the length and breadth respectively of its base. Then the solid content of the parallelopiped is the product of these quantities, or zxly =y(, y) Ix. $y. J-jCL Xq, X-i) ttn XIH) yVvV% y-. be corresponding successive values of the co-ordinates, and Ix, ly, the common differences of the successive values of x and y respectively. Then it may be seen that the solid Ac contains parallelopipeds, of which (reckoning them in rows parallel to ab) the solid contents ...