. Differential and integral calculus, an introductory course for colleges and engineering schools. 1 < ^^ < —^, whence , v , < r OPP < .. OPM (c) 1 = ^opn = ^Zopn Now OPN = ip2Ad = iP2dd and OPM = \ (p + Ap) OPM (p + Ap)2OPA^ p2 and lim OPN -ft, (,+*£) -1. n=oo \ p / OPP Consequently, by reason of (c), lim npAr = 1, and, by Duhamels Theorem each OPP in (b) may be replaced by the corresponding OPAf, that is, by \ p2dd, so that we have y y area OBC = lim Y J p2 dd = \ lim V p2 d0. n=oo o n=oo o Finally, p2 is a function of 6, and we may therefore invoke ourfundamental theore
. Differential and integral calculus, an introductory course for colleges and engineering schools. 1 < ^^ < —^, whence , v , < r OPP < .. OPM (c) 1 = ^opn = ^Zopn Now OPN = ip2Ad = iP2dd and OPM = \ (p + Ap) OPM (p + Ap)2OPA^ p2 and lim OPN -ft, (,+*£) -1. n=oo \ p / OPP Consequently, by reason of (c), lim npAr = 1, and, by Duhamels Theorem each OPP in (b) may be replaced by the corresponding OPAf, that is, by \ p2dd, so that we have y y area OBC = lim Y J p2 dd = \ lim V p2 d0. n=oo o n=oo o Finally, p2 is a function of 6, and we may therefore invoke ourfundamental theorem of Art. 169, and thus get the formula as in Arts. 165 and 170. Let us apply our fundamental theorem to obtain the formulafor the length of an arc of a curve s = f Vl +y*dx. s area OPC = \ I P2dd, §173 THE DEFINITE INTEGRAL 251 Let s be the length of the arc AB, and let this arc be dividedinto n small arcs whose lengths are denoted by As0, Asi, As2. . .Then, whatever value n may Yhave, 6 s = 5) As, a and when n is increased indefi-nitely by repeated subdivision ofthe small arcs, we have. (d) = lim ]T As. We have the well-known relations As _ ds_ ^_^s. AAl=0Ax dx Ax ~ dx where e is infinitesimal. Hence As 1 . dx . .. As -J- = l + €-r-, and lim-T-=ds ds ds + edx, 1. Therefore, by our theorem of infinitesimals, each As in (d) may bereplaced by the corresponding ds, and we have b b s = lim V ds = lim TV1+ y2 dx. „==o 0 n^oc^ But Vl + 2/2 is a function of x, and therefore to the last sum-mation our fundamental theorem applies, so that s= f VI + y2dx, and this is the formula It has been proved here incidentally that .. increment of the function _differential of the function Query. What does J) ds represent geometrically? 252 INTEGRAL CALCULUS §173 Problem 1. Let ABCD be an area bounded by a circle, its involute, and two positions AB and CD of thegenerating line. Show that \a2B2ddmay be taken for the element of area,and that ABCD = ^(d2*-eS).6 Problem 2. There
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