Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 404: 65



Work Step by Step

Since the volume is given by $$V=\pi \int_{a}^{b} f(x)^{2} d x$$ Then \begin{align*} V&=\pi \int_{0}^{\pi} \sin ^{2} x d x\\ &=\frac{\pi}{2} \int_{0}^{\pi}[1-\cos 2x]d x\\ &=\left.\pi\left(\frac{x}{2}-\frac{\sin 2 x}{4}\right)\right|_{0} ^{\pi}\\ &=\pi\left(\frac{\pi}{2}\right)\\ &=\frac{\pi^{2}}{2} \end{align*}
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