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lecture15 - Lecture 15 Triple Integrals June 4 2009 Lecture...

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Lecture 15: Triple Integrals June 4, 2009 Lecture 15: Triple Integrals
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Objectives 1 Compute triple integrals over rectangular boxes. 2 Compute triple integrals over more general regions. 3 Use triple integrals to compute volumes. 4 Use triple integrals to compute centers of mass. Lecture 15: Triple Integrals
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Triple integrals over boxes Suppose R = [ a , b ] × [ c , d ] × [ r , s ] is a rectangular region and f = f ( x , y , z ). ZZZ R f ( x , y , z ) dxdydz = Z s r Z d c Z b a f ( x , y , z ) dxdydz Lecture 15: Triple Integrals
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Triple integrals over boxes Suppose R = [ a , b ] × [ c , d ] × [ r , s ] is a rectangular region and f = f ( x , y , z ). ZZZ R f ( x , y , z ) dxdydz = Z s r Z d c Z b a f ( x , y , z ) dxdydz Fubini’s Theorem: We can integrate in any order. Lecture 15: Triple Integrals
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Triple integrals over general regions 1 Type I R = { ( x , y , z ) | ( x , y ) Du 1 ( x , y ) z u 2 ( x , y ) } 2 Type II R = { ( x , y , z ) | ( y , z ) Du 1 ( y , z ) x u 2 ( y , z ) } 3 Type III R = { ( x , y , z ) | ( x , z ) Du 1 ( x , z ) y u 2 ( x , z ) } Lecture 15: Triple Integrals
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Finding volumes with triple integrals
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