13_7 - 13.7 Triple Integrals in Cartesian Coordinates Let B...

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Unformatted text preview: 13.7 Triple Integrals in Cartesian Coordinates Let B = { ( x,y,z ) : a x b, c y d, e z g } RRR B f ( x,y,z ) dV = lim k P k n k =1 f ( x k , y k , z k ) V k = R b a R d c R g e f ( x,y,z ) dz dy dx 1 Example Let B = { ( x,y,z ) : 0 x 2 , 1 y 3 , 2 z 4 } Evaluate RRR B 4 x 2 yz dV . RRR B 4 x 2 yz dV = R 2 R 3 1 R 4 2 4 x 2 yz dz dy dx = R 2 R 3 1 (2 x 2 yz 2 ) / 4 2 dy dx = R 2 R 3 1 24 x 2 y / dy dx = R 2 12 x 2 y 2 / 3 1 dx = R 2 96 x 2 dx = 32 x 3 / 2 = 256 2 Let S be a z-simple set, and let S xy be its projection in the xy-plane, i.e. S = { ( x,y,z ) : 1 ( x,y ) z 2 ( x,y ) , ( x,y ) S xy } Then ZZZ S f ( x,y,z ) dV = ZZ S x,y " Z 2 ( x,y ) 1 ( x,y ) f ( x,y,z ) dz # dA If S xy is y-simple, i.e. S x,y = { ( x,y ) : 1 ( x ) y 2 ( x ) , a x b } Then RRR S f ( x,y,z ) dV = R b a R 2 ( x ) 1 ( x ) R 2 ( x,y ) 1 ( x,y ) f ( x,y,z ) dz dy dx Similarly, we can define...
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13_7 - 13.7 Triple Integrals in Cartesian Coordinates Let B...

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