University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 10 - Section 10.5 - Areas and Lengths in Polar Coordinates - Exercises - Page 585: 5



Work Step by Step

The area of a shaded region can be found as: $A=\dfrac{1}{2}\int_p^q r^2 d \theta$ Here, we have $A=\int_{-\pi/4}^{\pi/4} \dfrac{1}{2}\cos^2 2 \theta d \theta$ $(2)\dfrac{1}{2} \int_{-\pi/4}^{\pi/4} (\dfrac{1+\cos 4 \theta}{2})d \theta=\dfrac{1}{2}[ \dfrac{\pi}{4}+\dfrac{ \sin 4 \theta}{\theta}]_{0}^{\pi/4} $ Thus, $A=\dfrac{\pi}{8}$
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