Calculus (3rd Edition)

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

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.4 Area and Arc Length in Polar - Exercises - Page 625: 36


$$\int_{0}^{\pi}\sqrt{\sin^2\theta\cos^2\theta+\cos^22\theta} d \theta\\ =\int_{0}^{\pi}\sqrt{\frac{1}{4}\sin^22\theta+\cos^22\theta} d\theta $$

Work Step by Step

Since $r=f(\theta)=\sin\theta\cos\theta$, then $f'(\theta)=\cos^2\theta-\sin^2\theta=\cos2\theta$ Thus, the length is given by $$ \text { Length }=\int_{0}^{\pi} \sqrt{f(\theta)^{2}+f^{\prime}(\theta)^{2}} d \theta\\ =\int_{0}^{\pi}\sqrt{\sin^2\theta\cos^2\theta+\cos^22\theta} d \theta\\ =\int_{0}^{\pi}\sqrt{\frac{1}{4}\sin^22\theta+\cos^22\theta} d\theta $$
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