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 909: 21


$$I_x=\frac{M}{3}\left(b^{2}+c^{2}\right),\ \ I_{y}=\frac{M}{3}\left(a^{2}+c^{2}\right),\ \ I_{z}=\frac{M}{3}\left(a^{2}+b^{2}\right),$$

Work Step by Step

Since \begin{align*} I_{x}&=\int_{0}^{a} \int_{0}^{b} \int_{0}^{c}\left(y^{2}+z^{2}\right) d z d y d x\\ &=\int_{0}^{a} \int_{0}^{b}\left(c y^{2}+\frac{c^{3}}{3}\right) d y d x\\ &=\int_{0}^{a}\left(\frac{c b^{3}}{3}+\frac{c^{3} b}{3}\right) d x\\ &=\frac{a b c\left(b^{2}+c^{2}\right)}{3}\\ &=\frac{M}{3}\left(b^{2}+c^{2}\right) \end{align*} where $M=a b c ; $ and, by symmetry $$I_{y}=\frac{M}{3}\left(a^{2}+c^{2}\right),\ \ I_{z}=\frac{M}{3}\left(a^{2}+b^{2}\right)$$
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