Advanced Calculus fi..

262 Advanced Calculus, Fifth Edition *ITHEOREM N Let f (x , y) be continuous on the set G : a 5 x 5 b, y (x) 5y 5 y2(x), where yl(x) and y2(x) are continuous for a 5 x 5 b and yl(x) < y2(x)for a < x < b. Then the double integral of f over G exists and equals the iteratedintegralProof. As in Section 4.3, we subdivide the region by parallels to the axes: Linesx = X, = const for a = xo < xl < ... < x, = b, lines y = y, = const foryo < y~ < .. . < y,. Here yo must be less than or equal to the minimum of yl(x),and y, must be greater than or equal to the maximum of y2(x). These lines formrectangles, and we let h be the mesh of the subdivision, the largest diagonal of therectangles.By Theorem M the iterated integral (4.98) exists and has a value c. We let g(x)denote the inner integral, as in (4.96), so that g(x) is continuous. Nowby the Mean Value theorem, with xi-, 5 xf 5 xi and Aix xi - for i = ,1, . . . , n. Similarly,Here we use the values y,, to subdivide the interval yl(x:) 5 y 5 y2(x:), lettingy, I be the first of the values yo, yl, ... to exceed yl (x:) and y,,,,-1 be the last to fallbelow Y~(x:). We also write%(see Fig. 4.16). We writeY~O = YI(x:), Ytm, = Y~(x:)?13 Figure 4.16 Formation of sum (4.101).