Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 11: Parametric Equations and Polar Coordinates - Section 11.5 - Areas and Lengths in Polar Coordinates - Exercises 11.5 - Page 671: 29



Work Step by Step

Let us consider that $r=f(\theta)$ Since $x=r \cos \theta$ then $ x=f(\theta) \cos \theta \\ y=r \sin \theta \implies y=f(\theta) \sin \theta$ Use formula for the length curve: $L=\int_{a}^{b}\sqrt{(\dfrac{dx}{dt})^2+(\dfrac{dy}{dt})^2}dt$ $L=\int_{a}^{b} \sqrt{(f'(\theta) (\cos \theta)-f(\theta) (\sin \theta)^2+(f'(\theta) (\sin \theta)+f(\theta) (\cos \theta)^2}dt$ or, $L=\int_{a}^{b}\sqrt{[f'(\theta)]^2 +f(\theta)^2}d\theta$ Hence, the length of the curve can be written as: $L=\int_{a}^{b}\sqrt{r^2+(\dfrac{dr}{d\theta})^2}d\theta$
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