Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 15 - Multiple Integrals - 15.7 Triple-Integrals in Cylindrical Coordinates - 15.7 Exercises - Page 1084: 21

Answer

$\dfrac{2\pi}{5}$

Work Step by Step

$I=\iiint_Ex^2dV=\int_0^{2\pi} \int_{0}^{1}\int_{0}^{2r} x^2 \times (r dz dr d\theta) $ $=\int_0^{2\pi} \int_{0}^{1}\int_{0}^{2r} (r\cos \theta)^2 \times (r dz dr d\theta) $ $=\int_0^{2\pi} \int_{0}^{1}[r^3 \times \cos^2 \theta(z)]_{0}^{2r} dr d\theta$ $=\int_0^{2\pi} \int_{0}^{1}2 \times r^4 \times \cos^2 \theta dr d\theta$ $=\int_0^{2\pi}(\dfrac{2}{5}) \times \cos^2 \theta d\theta$ $=0.2 \times \int_0^{2\pi} (1+\cos 2 \theta) d\theta$ $=0.2 \times \int_0^{2\pi} [\theta+\dfrac{\sin 2 \theta}{2})_0^{2\pi}$ $=\dfrac{2\pi}{5}$
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