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 610: 5



Work Step by Step

\begin{aligned} s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\ s &=\int_{1}^{4} \sqrt{36 t^{2}+144 t^{4}} d t \\ &=\int_{1}^{4} \sqrt{36 t^{2}\left(1+4 t^{2}\right)} d t \\ &=\frac{6}{8} \int_{1}^{4} 8 t\left(1+4 t^{2}\right)^{1 / 2} d t\\ &= \frac{3}{4}\frac{2}{3}(1+4t^2)^{3/2}\bigg|_{1}^{4}\\ &= \frac{5\left(13\sqrt{65}-\sqrt{5}\right)}{2} \end{aligned}
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