First Principles Of The Differential And Integral Calculus PDF Download

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1836 edition. Excerpt: ...of the surface reckoned from A, the origin of the abscissas, than it is the differential of any other surface KPML reckoned from a fixed and determinate point K; since we have equally PpmM= Kp mL--KPML = d (KPML). When we integrate, therefore, we have no right to refer the integral given directly by the calculus to the space APM, rather than to any other space KPLM, which differs from it by a determinate and constant space KAL. We must therefore add to the integral found by the calculus, a constant quantity expressing that by which the space proposed to be determined differs from that given directly by the calculus. It will be seen in the following examples, how this constant quantity is determined. Let us, in the first place, find the expression /or the space P p mM. Calling AP, x; PM, y; we have Pp = dx, pm = y--di/. The surface of the trapezium P p m M is (Geo. 178) But to indicate that d y and d x are infinitely small, we must reject--i---, which is infinitely small, compared with ydx; we At have therefore y d x as the general expression of the differential of the /element of the surface of any curve. In order to apply this formula to a proposed surface, whose equation is given, we must deduce from this equation the value of y, which we substitute in the formula y d x, we then have a quantity in terms of x and d x, which, when it can be integrated by the preceding rules, will give, with the addition of a constant quantity, the expression of the surface of this curve, reckoned from any point we please. We have then only to determine the constant quantity, which is done by expressing from what point we choose to estimate the surface. We shall now illustrate this theory by examples. Let us take, for the first example, the common parabola, .