MULTIPLE INTEGRATION - Spring 2010-11 MULTIPLE INTEGRATION...

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Spring 2010-11 MULTIPLE INTEGRATION The integrals of functions of more than one variable are known as multiple integrals and are evaluated by a process involving iterated integrals. Partial Integration: The process in which the integration is performed with respect to one variable treating the other variable(s) as constant is called partial integration. Iterated Integral: A definite integral which is evaluated stage by stage using partial integration is called an iterated ( successive or repeated ) integral. 1. Evaluate the following iterated integrals: (a) ∫∫ + 1 0 2 0 ) 2 ( dx dy x (b) ∫∫ + 4 2 3 0 ) ( dxdy y x (c) ∫∫ + 1 0 1 0 ) ( 2 2 dydx xye y x (d) ∫ ∫ + 1 0 0 2 ) ( x dx dy y y (e) ∫∫ + 2 1 1 ) ( 1 1 y dy dx y x (f) ∫ ∫ e y dy dx xy 1 2 / 1 0 ) cos( π 2. Evaluate the following iterated integrals: (a) ∫∫∫ 2 0 1 0 2 1 2 dxdydz yz x (b) ∫∫ ∫ - + + 2 1 1 0 1 1 2 2 2 ) ( dxdydz z y x (c) ∫ ∫∫ π π φ θ θ 2 0 0 0 2 sin a d drd r (d) ∫∫ ∫ - 1 0 0 0 x y x dzdydx x (e) ∫ ∫ - - 1 0 1 0 1 0 2 x y dzdydx z (f) ∫∫ ∫ + 2 1 0 3 0 2 2 z x dydxdz y x x
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