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 - 12.2 Arc Length and Speed - Exercises - Page 611: 13


$$\frac{1}{2}\left(\sqrt{5}+\frac{1}{2}\ln \left(2+\sqrt{5}\right)\right)$$

Work Step by Step

\begin{aligned} s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\ &=\int_{0}^{1} \sqrt{ 1+4t^2} d t \\ &= t \sqrt{1+4t^{2}}+\frac{1}{2} \ln(2t+\sqrt{1+4t^{2}})\bigg|_{0}^{1}\\ &= \frac{1}{2}\left(\sqrt{5}+\frac{1}{2}\ln \left(2+\sqrt{5}\right)\right) \end{aligned}
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