Calculus: Early Transcendentals 8th Edition

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

Chapter 7 - Section 7.2 - Trigonometric Integrals - 7.2 Exercises - Page 485: 62



Work Step by Step

$\displaystyle{A(x)=\pi\left(\sin^2x\right)^2}\\$ $\displaystyle{V=\int_0^\pi A(x)\ dx}\\ \displaystyle{V=\int_0^\pi \pi\left(\sin^2x\right)^2\ dx}\\ \displaystyle{V=\pi\int_0^\pi \left(\sin^2x\right)^2 \ dx}\\ \displaystyle{V=\pi\int_0^\pi \left(\frac{1-\cos 2 x}{2}\right)^{2} dx}\\ \displaystyle{V=\frac{\pi}{4} \int_0^\pi \left(1-2 \cos 2 x+\cos ^{2} 2 x\right) dx}\\ \displaystyle{V=\frac{\pi}{4} \int_0^\pi1-2 \cos 2 x+\frac{1}{2}\left(1+\cos 4x\right)\ dx}\\ \displaystyle{V=\frac{\pi}{4} \int_0^\pi\frac{1}{2}\cos 4x-2 \cos 2 x+\frac{3}{2}\ dx}\\ \displaystyle{V=\frac{\pi}{4} \left[\frac{1}{8}\sin 4x-\sin 2 x+\frac{3}{2}x\right]_0^\pi}\\ \displaystyle{V=\frac{3\pi^2}{8}}$
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