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: 28


$\sqrt 2 (e^{2\pi}-1).$

Work Step by Step

Since $r=f(\theta)=e^\theta$, then $f'(\theta)=e^\theta$. Thus, the length is given by \begin{align*} \text { Length }&=\int_{0}^{2\pi} \sqrt{f(\theta)^{2}+f^{\prime}(\theta)^{2}} d \theta\\ &=\int_{0}^{2\pi}\sqrt{e^{2\theta}+e^{2\theta}} d \theta\\ &=\int_{0}^{2\pi}\sqrt 2 e^\theta d \theta\\ &=\sqrt 2 e^\theta |_{0}^{2\pi}\\ &=\sqrt 2 (e^{2\pi}-1). \end{align*}
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