Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 321 APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH 251EXAMPLE Find the area d the region bounded by the parahola y = x* and the tine y = x + 2 (sec Fig 32-7).The limits of in1egralic.w U and h in (32.1) musi be rhe xcoordinatcs of the inkmc1ion points P and Q.respectively. These arc found by solving simultamously the equations of the cum y = xz and y = x + 2. Thus,x*=x+2 or xz-x-2-0 or (~-2Nx+l)=Owhm%x D -1 WVJX = b 2. Th\&,4.v32.3 ARC LENGTHConsider a differentiable (not just continuous) functionfon a closed interval [U, b]. The graph offis a curve running from (o.f(u)) to (b,f(b)). We shall find 3 formula for the length L of this curveDividc [U, h] into n equal parts, each of lmgrh Ax. To each xt in this subdivision corresponds the-point PAX,, /(xi)) on the curve (see Fig. 32-8). For large n, the sum POP, + v,-+ .-. + Pm- sE- P,- rYi of the lengths of the line segments P,- ,P, is an approximation to the length of the curve.Now, by the distance formula (2.1).p4 - 1 = J(xi - xi - a)* + (/(xi) -j(xi-But x4 - xi-l = Ax; also. by the mean-value theorem (Theorem 17.2).I(x3 -f(x4- 1) = Ixr - xi- a).f’(xF) = 4W.f’(%’)for some xr in (x,- xi). Hence.f,- *pi = J(W’ + (Wmm2 = J{ 1 + ~Y(XW)(WZ= J- rn = J- Ax