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13.2 Iterated Integrals Suppose that f (x, y) is an integrable over the rectangle R = [a, b] × [c, d]. ˆ Partial integration b of f with respect to x: f (x, y) dx a means that y is held xed and f (x, y) is integrated with respect to x from x = a to x = b. ˆ Partial integration d of f with respect to y : f (x, y) dy c means that x is held xed and f (x, y) is integrated with respect to y from y = c to y = d. Example 1. ˆ 2 (a) (x + 3y 2 ) dx 0 ˆ (b) 3 exy dy 1 Iterated integrals : ˆ dˆ b ˆ d ˆ b f (x, y) dxdy := c a f (x, y) dx dy c a means that we rst integrate with respect to x from a to b and then with respect to y from c to d. ˆ bˆ ˆ b ˆ d f (x, y) dydx := a c d f (x, y) dy dx a c means that we rst integrate with respect to y from c to d and then with respect to x from a to b. Example 2. Evaluate the iterated integrals: (a) ˆ 4ˆ 1 (b) √ x y dxdy 4 √ x y dydx 0 ˆ 1ˆ 0 1 1 Fubini's Theorem ¨ . If f is continuous on the rectangle R = [a, b] × [c, d], then ˆ dˆ b f (x, y) dA = R ˆ bˆ f (x, y) dxdy = c Example 3. Evaluate a x cos(xy) dA R where R = [−π/2, π/2] × [1, 5]. f (x, y) dydx a ¨ d c Example 4. Find the volume of the solid S lying under the paraboloid z = x2 + y 2 and above the rectangle R = [−2, 2] × [−3, 3]. Fact then : If g(x) and h(y) are continuous functions of one variable and R = [a, b] × [c, d], ¨ ˆ ˆ b g(x)h(y) dA = g(x) dx a R Example 5. If R = [0, ln 2] × [0, ln 5], nd ¨ e2x−y dA R d h(y) dy c