Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 461: 91



Work Step by Step

The volume is given by \begin{align*} V&=2 \pi \int_{0}^{\infty}xf(x)dx\\ &=2 \pi \int_{0}^{\infty} \frac{x}{\left(x^{2}+1\right)^{2}} d x\\ &= 2 \pi\lim _{R \rightarrow \infty} \int_{0}^{R} \frac{x d x}{\left(x^{2}+1\right)^{2}}\\ &=2 \pi \lim _{R \rightarrow \infty} \frac{-1}{2(x^2+1)} \bigg|_{0}^{R}\\ &=2 \pi \lim _{R \rightarrow \infty} \frac{1}{2}\left(1-\frac{1}{R^{2}+1}\right)\\ &=\pi \end{align*}
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