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: 35


$$ \text { The length } =\int_{0}^{2\pi} \sin^2\theta\sqrt{1 +8\cos^2\theta} d \theta. $$

Work Step by Step

Since $r=f(\theta)= \sin^3\theta$, then $f'(\theta)=3\sin^2\theta\cos\theta$ The length is given by $$ \text { The length }=\int_{0}^{2\pi} \sqrt{f(\theta)^{2}+f^{\prime}(\theta)^{2}} d \theta\\ =\int_{0}^{2\pi} \sqrt{\sin^6\theta +9\sin^4\theta\cos^2\theta} d \theta\\ =\int_{0}^{2\pi} \sin^2\theta\sqrt{\sin^2\theta +9\cos^2\theta} d \theta\\ =\int_{0}^{2\pi} \sin^2\theta\sqrt{1 +8\cos^2\theta} d \theta. $$
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