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16.4 Ex. 8

# 16.4 Ex. 8 - S E C T I O N 16.4 SOLUTION Integration in...

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S E C T I O N 16.4 Integration in Polar, Cylindrical, and Spherical Coordinates (ET Section 15.4) 941 SOLUTION Step 1. Express W in cylindrical coordinates. W is bounded above by the plane z = x and below by z = 0, therefore 0 z x , in particular x 0. Hence, W projects onto the semicircle D in the xy -plane of radius 3, where x 0. x D y 3 3 In polar coordinates, D : − π 2 θ π 2 , 0 r 3 The upper surface is z = x = r cos θ and the lower surface is z = 0. Therefore, W : − π 2 θ π 2 , 0 r 3 , 0 z r cos θ Step 2. Set up an integral in cylindrical coordinates and evaluate. The volume of W is the triple integral W 1 dV . Using change of variables in cylindrical coordinates gives W 1 dV = π / 2 π / 2 3 0 r cos θ 0 r dz dr d θ = π / 2 π / 2 3 0 rz r cos θ z = 0 dr d θ = π / 2 π / 2 3 0 r 2 cos θ dr d θ = π / 2 π / 2 r 3 3 cos θ 3 r = 0 d θ = π / 2 π / 2 9 cos θ d θ = 9 sin θ π / 2 π / 2 = 9 sin π 2 sin π 2 = 18 8. Let W be the region above the sphere x 2 + y 2 + z 2 =

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• Winter '08
• Rogawski
• Coordinate system, Spherical coordinate system, Polar coordinate system, Cylindrical coordinate system, cylindrical coordinates, dr d

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16.4 Ex. 8 - S E C T I O N 16.4 SOLUTION Integration in...

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