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: 9


$$ \bar{x}=-1\ \ \text{ and }\ \ \bar{y}=\frac{1}{4}$$

Work Step by Step

Since \begin{align*} M&=\int_{-\infty}^{0} \int_{0}^{e^{x}} d y d x\\ &=\int_{-\infty}^{0} e^{x} d x=\lim _{b \rightarrow-\infty} \int_{b}^{0} e^{x} d x\\ &=1-\lim _{b \rightarrow-\infty} e^{b}=1\\ M_{y}&=\int_{-\infty}^{0} \int_{0}^{e^{x}} x d y d x\\ &=\int_{-\infty}^{0} x e^{x} d x\\ &=\lim _{b \rightarrow-\infty} \int_{b}^{0} x e^{x} d x\\ &=\lim _{b \rightarrow-\infty}\left[x e^{x}-e^{x}\right]_{b}^{0}\\ &=-1-\lim _{b \rightarrow-\infty}\left(b e^{b}-e^{b}\right)\\ &=-1\\ M_{x}&=\int_{-\infty}^{0} \int_{0}^{e^{x}} y d y d x\\ &=\frac{1}{2} \int_{-\infty}^{0} e^{2 x} d x\\ &=\frac{1}{2} \lim _{b \rightarrow-\infty} \int_{b}^{0} e^{2 x} d x\\ &=\frac{1}{4} \end{align*} Then $$ \bar{x}=-1\ \ \text{ and }\ \ \bar{y}=\frac{1}{4}$$
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