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Calculus II Notes 16.2

Calculus II Notes 16.2 - Calculus II Stewart Dr Berg Spring...

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Unformatted text preview: Calculus II- Stewart Dr. Berg Spring 2010 Page 1 16.2 16.2 Iterated Integrals If we integrate a function f ( x , y ) with respect to y (treating x as a constant), we get a function of x : A ( x ) = f ( x , y ) c d ∫ dy . If we integrate A ( x ) = f ( x , y ) c d ∫ dy with respect to x , we get an iterated integral : A ( x ) a b ∫ dx = f ( x , y ) c d ∫ dy ( ) a b ∫ dx . Is f ( x , y ) c d ∫ dy ( ) a b ∫ dx = f ( x , y ) a b ∫ dx ( ) c d ∫ dy ? Example A Find the integral of f ( x , y ) = x 2 − xy over the rectangle R = [0,2] × [0,2] in both orders. Solution : x 2 − xy ( ) 2 ∫ 2 ∫ dx dy = x 3 3 − x 2 y 2 2 ∫ 2 dy = 2 3 3 − 2 2 y 2 − 2 ∫ 2 dy = 8 3 − 2 y 2 ∫ dy = 8 3 y − y 2 2 = 16 3 − 4 = 4 3 . Also x 2 − xy ( ) 2 ∫ 2 ∫ dy dx = x 2 y − xy 2 2 2 ∫ 2 dx = 2 x 2 − 2 x ( ) 2 ∫ dx = 2 x 3 3 −...
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Calculus II Notes 16.2 - Calculus II Stewart Dr Berg Spring...

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