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Lecture 11, October 17 • Double integrals. The double integral of f (x, y) over a region R in the xy-plane is deﬁned in terms of Riemann sums as ∫∫ n ∑ f (x∗k , yk∗ ) ∆Ak . f (x, y) dA = lim n→∞ R k=1 If f is positive, then this expression gives the volume of the solid that lies below the graph of f and above the region R in the xy-plane. When R = [a, b] × [c, d], one has ∫∫ ∫ d∫ b ∫ b∫ d f (x, y) dA = f (x, y) dx dy = f (x, y) dy dx. c R a a c ..................................................................................... Example 1. Consider f (x, y) = x2 y over the rectangle R = [0, 2] × [0, 1]. Then ∫∫ ∫ 1∫ 2 f (x, y) dA = x2 y dx dy 0 R 0 and we can focus on the inner integral ﬁrst. Integrating with respect to x, we get [ 3 ]2 ∫ 2 ∫ 2 8y x 2 2 = x y dx = y x dx = y 3 x=0 3 0 0 and so the double integral is ∫∫ ∫ f (x, y) dA = 0 R 1 [ 2 ]1 4y 4 8y = . dy = 3 3 y=0 3 Alternatively, one may reach the same answer by writing ∫∫ ∫ 2∫ 1 f (x, y) dA = x2 y dy dx R 0 0 and by integrating with respect to y ﬁrst. This approach gives [ 2 ]1 ∫ 1 ∫ 1 x2 2 2 2 y x y dy = x y dy = x = 2 y=0 2 0 0 for the inner integral, so the double integral is equal to [ 3 ]2 ∫∫ ∫ 2 2 x x 4 f (x, y) dA = dx = = . 6 x=0 3 0 2 R