This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 xyplane 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...
View
Full
Document
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.
 Winter '08
 Rogawski

Click to edit the document details