16.4 Ex. 8

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

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Unformatted text preview: 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 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 : 2 2 , r 3 The upper surface is z = x = r cos and the lower surface is z = 0. Therefore, W : 2 2 , r 3 , z r cos Step 2. Set up an integral in cylindrical coordinates and evaluate. The volume of W is the triple integral ZZZ W 1 dV . Using change of variables in cylindrical coordinates gives ZZZ W 1 dV = Z / 2 / 2 Z 3 Z r cos r dz dr d = Z / 2 / 2 Z 3 rz r cos z = dr d = Z / 2 / 2 Z 3 r 2 cos dr d = Z / 2 / 2 r 3 3 cos 3 r = d = Z / 2 / 2 9 cos...
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This homework help was uploaded on 04/13/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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

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