Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 10 - Parametric Equations and Polar Coordinates - 10.4 Areas and Lengths in Polar Coordinates - 10.4 Exercises - Page 712: 2



Work Step by Step

$A=\int ^{\pi /6}_{0}\dfrac {1}{2}r^{2}d\theta =\int ^{\dfrac {\pi }{6}}_{0}\dfrac {1}{2}\cos ^{2}\theta d\theta =\int ^{\dfrac {\pi }{6}}_{0}\dfrac {1}{2}\times \dfrac {1+\cos 2\theta }{2}d\theta =\dfrac {1}{4}\left[ \theta +\dfrac {1}{2}\sin 2\theta \right] ^{\pi /6}_{0}=\dfrac {1}{4}\left( \dfrac {\pi }{6}+\dfrac {\sin \dfrac {\pi }{3}}{2}\right) =\dfrac {1}{8}\left( \dfrac {\pi }{3}+\dfrac {\sqrt {3}}{2}\right) \approx 0.24$
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