. Differential and integral calculus, an introductory course for colleges and engineering schools. TE INTEGRAL lim V/(x)Ax = f f(x)dx. 243 Not only, then, has the sum of infinitesimal products]^ f(x)Ax a a finite and determinate limit, but this limit is a definite integral. These results are of such importance in the Integral Calculus that we restate them in the Fundamental Theorem. Let f(x) be real, single-valued, and continuous throughout an interval a = x = b, and let this interval be subdivided into n smaller intervals by the points Xi, x2, . . xn-i, b and let the sum of products be formed
. Differential and integral calculus, an introductory course for colleges and engineering schools. TE INTEGRAL lim V/(x)Ax = f f(x)dx. 243 Not only, then, has the sum of infinitesimal products]^ f(x)Ax a a finite and determinate limit, but this limit is a definite integral. These results are of such importance in the Integral Calculus that we restate them in the Fundamental Theorem. Let f(x) be real, single-valued, and continuous throughout an interval a = x = b, and let this interval be subdivided into n smaller intervals by the points Xi, x2, . . xn-i, b and let the sum of products be formed, ^f(x)Ax. Now let n, the a number of subintervals, be increased indefinitely by repeated subdi-vision of each subinterval, the process of division being carried on in such a way that each subinterval has the limit 0. Then, although b each product f(x)dx has the limit 0, the sum ^f(x)dx has a limit a which is the definite integral, f(x) dx. That is, x lim *%f{x)dx= I f(x)dx. n=oo a Ua It is evident from our defini-tion and from the accompany-ing figures that Jf(x) dx = — I f(x) dx,a Jb and. fbf(x)dx+ Pf(x)dx+ fdf(x)dx= fdf(x)dx, *J a Ub %Jc *Ja and that these hold true whatever be the order of the limits, a, b,c, d. Problem. Show how the proof of the fundamental theorem dependsupon the continuity of fix). 244 INTEGRAL CALCULUS §170 We have already stated that, although the foregoing proof ofthe fundamental theorem is geometric, the theorem itself is apurely analytic one. This means that the theorem holds truewhen the values of f(x) are something else than the ordinates ofa curve, and when therefore the limit of the sum of products issomething else than an area. We shall have many examples ofthis. It may be remarked here that this property of being the limitof a sum of products of the form f(x)dx could be taken as thedefinition of the definite integral. Indeed, to the originators ofthe Calculus the definite integral first presented itself as the limitof such a sum of products
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