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 908: 11


$$M=\frac{8}{15},\ \ I_x=\frac{64}{105}$$

Work Step by Step

Since \begin{align*} M&=\int_{0}^{2} \int_{-y}^{y-y^{2}}(x+y) d x d y\\ &=\int_{0}^{2}\left[\frac{x^{2}}{2}+x y\right]_{-y}^{y-y^{2}} d y\\ &=\int_{0}^{2}\left(\frac{y^{4}}{2}-2 y^{3}+2 y^{2}\right) d y\\ &=\left[\frac{y^{5}}{10}-\frac{y^{4}}{2}+\frac{2 y^{3}}{3}\right]_{0}^{2}\\ &=\frac{8}{15} \end{align*} and \begin{align*} I_{x}&=\int_{0}^{2} \int_{-y}^{y-y^{2}} y^{2}(x+y) d x d y\\ &=\int_{0}^{2}\left[\frac{x^{2} y^{2}}{2}+x y^{3}\right]_{-y}^{y-y^{2}} d y\\ &=\int_{0}^{2}\left(\frac{y^{6}}{2}-2 y^{5}+2 y^{4}\right) d y\\ &=\frac{64}{105} \end{align*}
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