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 15.6 - Moments and Centers of Mass - Exercises 15.6 - Page 910: 33



Work Step by Step

\begin{align*} M&=\int_{-1}^{1} \int_{z-1}^{1-z} \int_{0}^{\sqrt{z}}(2 y+5) d y d x d z\\ &=\int_{0}^{1} \int_{z-1}^{1-z}(z+5 \sqrt{z}) d x d z\\ &=\int_{0}^{1} 2(z+5 \sqrt{z})(1-z) d z\\ &=2 \int_{0}^{1}\left(5 z^{1 / 2}+z-5 z^{3 / 2}-z^{2}\right) d z\\ &=2\left[\frac{10}{3} z^{3 / 2}+\frac{1}{2} z^{2}-2 z^{5 / 2}-\frac{1}{3} z^{3}\right]_{0}^{1}\\ &=2\left(\frac{9}{3}-\frac{3}{2}\right)=3 \end{align*}
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