Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.1 - Integration by Parts - 7.1 Exercises - Page 477: 63


$\displaystyle{V=2\pi e}$

Work Step by Step

$\displaystyle{V=\int_{-1}^0 \left(2\pi (1-x)\right)\left(e^{-x}\right)\ dx}\\ \displaystyle{V=2\pi\int_{-1}^0 e^{-x}-xe^{-x}\ dx}\\ \displaystyle{V=2\pi\int_{-1}^0 e^{-x}\ dx-2\pi\int_{-1}^0xe^{-x}\ dx}\\$ $\displaystyle \left[\begin{array}{ll} u=x & dv=e^{-x} \\ & \\ du=1 & v=-e^{-x} \end{array}\right]$ Integration by parts $\displaystyle{V=2\pi \left([-e^{-x}]_{-1}^0\right)-2\pi[-xe^{-x}]_{-1}^0+2\pi\int_{-1}^0-e^{-x}\ dx}\\ \displaystyle{V=2\pi \left(-1+e\right)-2\pi \left(-1\right)}\\ \displaystyle{V=2\pi e}$
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