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

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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→ 0 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
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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 0 R 3 1 R 4 2 4 x 2 yz dz dy dx = R 2 0 R 3 1 £ (2 x 2 yz 2 ) / 4 2 dy dx = R 2 0 R 3 1 £ 24 x 2 y / dy dx = R 2 0 £ 12 x 2 y 2 / 3 1 dx = R 2 0 96 x 2 dx = £ 32 x 3 / 2 0 = 256 2
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Let S be a z -simple set, and let S x y be its projection in the xy -plane, i.e. S = { ( x, y, z ) : ψ 1 ( x, y ) z ψ 2 ( x, y ) , ( x, y ) S x y } Then ZZZ S f ( x, y, z ) dV = ZZ S x,y " Z ψ 2 ( x,y ) ψ 1 ( x,y ) f ( x, y, z ) dz # dA If S x y is y -simple, i.e. S x,y = { ( x, y ) : φ 1 ( x ) y φ 2 ( x ) , a x b } Then RRR S f ( x, y, z ) dV = R b a R φ 2 ( x ) φ 1 ( x ) R ψ 2 ( x,y ) ψ 1 ( x,y ) f ( x, y, z ) dz dy dx Similarly, we can define x -simple and y -simple sets, and triple integrals over these sets. 3
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Example Evaluate R 1 0 R x +1 x R 3 y x x ( y +2 z ) dz dy dx .
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