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Page 1 Math 251-copyright Joe Kahlig, 15A Section 13.2: Iterated Integrals Definition: Suppose f is a function of two variables that is integrable over the rectangle R = [a, b] × [c, d]. Zb The partial integration of f with respect to x is f (x, y)dx and means that y is held fixed and f a is integrated with respect to x. Zd The partial integration of f with respect to y is f (x, y)dy and means that x is held fixed and f c is integrated with respect to y. Both of these integrals give an answer that is a function of the variable that is held fixed. Integrating this function with respect to the variable held fixed is called an iterated integral. Zb Zd Zd Zb f (x, y)dy dx f (x, y)dx dy a c c Example: Evaluate the following. Z3 Z2 (A) 0 1 Z2 Z3 (B) 1 6x2 y dydx 0 6x2 y dxdy a Page 2 Math 251-copyright Joe Kahlig, 15A Fubini’s Theorem: If f is continuous on the rectangle R = [a, b] × [c, d], then Zb Zd ZZ f (x, y)dA = f (x, y)dydx = a R Zd Zb c f (x, y)dxdy c a In the case where f (x, y) = g(x)h(y) then Zb Zd ZZ f (x, y)dA = R Zb g(x)h(y)dydx = a c Zd g(x)dx a h(y)dy c Example: Evaluate the double integral where R = {(x, y)|0 ≤ x ≤ 3, 1 ≤ y ≤ 5} = [0, 3] × [1, 5] ZZ (x + 3y 2 )dA R Example: If R = [1, 8] × [0, π], evaluate ZZ y cos(xy)dA R Math 251-copyright Joe Kahlig, 15A Page 3 Example: find the volume of the solid S that is bounded by the elliptic paraboloid z = 6x2 + y 2 + 1, the planes x = 3 and y = 4 and the three coordinate planes.