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 403: 7



Work Step by Step

\begin{aligned} \text{Area}&=\int_{0}^{\pi/2} \cos ^{3} x d x -\int_{\pi/2}^{3\pi/2} \cos ^{3} x d x \\ &=\int_{0}^{\pi/2} \cos ^{2} x \cos x d x -\int_{\pi/2}^{3\pi/2} \cos ^{2} x \cos x d x\\ &=\int_{0}^{\pi/2}\left(1-\sin ^{2} x\right) \cos x d x-\int_{\pi/2}^{3\pi/2}\left(1-\sin ^{2} x\right) \cos x d x \\ &=\int_{0}^{\pi/2}(\cos x d x)-\int_{0}^{\pi/2}\left(\sin ^{2} x\right) \cos x d x-\int_{\pi/2}^{3\pi/2}(\cos x d x)+\int_{\pi/2}^{3\pi/2}\left(\sin ^{2} x\right) \cos x d x \\ &=\sin x-\frac{1}{3} \sin ^{3} x\bigg|_{0}^{\pi/2}-\left(\sin x-\frac{1}{3} \sin ^{3} x\right)\bigg|_{\pi/2}^{3\pi/2}\\ &= 2 \end{aligned}
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