Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 15: Multiple Integrals - Section 15.7 - Triple Integrals in Cylindrical and Spherical Coordinates - Exercises 15.7 - Page 921: 67

Answer

$\bar x=\bar y=0$

Work Step by Step

We calculate the center of mass as follows: First calculate the mass: $M=4\int^{\pi/2}_0 \int^1_0 \int^r_0 $ dz r dr $ d\theta $ =$4\int^{\pi/2}_0 \int^1_0 r^2$ dr $ d\theta $ =$\frac{4}{3}\int^{\pi/2}_0 d\theta $ =$\frac{2\pi}{3}$ Now calculate the moment: $M_{xy}$= $\int^{2\pi}_0 \int^1_0 \int^r_0 $ z dz r dr $ d\theta $ =$\frac{1}{2}\int^{2\pi}_0 \int^1_0 r^3 dr d\theta $ =$\frac{1}{8} \int^{2\pi}_0 d\theta $ =$\frac{\pi}{4}$ And take the ratio: $\frac{M_{xy}}{M}=(\frac{\pi}{4})(\frac{3}{2\pi})=\frac{3}{8}$ Thus, we have (by symmetry): $\bar x=\bar y=0$
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