University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 14 - Section 14.4 - Double Integrals in Polar Form - Exercises - Page 777: 3


$\dfrac{\pi}{4} \leq \theta \leq \dfrac{3\pi}{4}$ and $0\leq r\leq \csc \theta$

Work Step by Step

Conversion of polar coordinates and Cartesian coordinates are as follows: a)$r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$ b) $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$ c) $x=r \cos \theta$ d) $y=r \sin \theta$ As $y=r \cos \theta =1\implies r=\csc \theta$ Therefore, the region described in polar coordinates is: $\dfrac{\pi}{4} \leq \theta \leq \dfrac{3\pi}{4}$ and $0\leq r\leq \csc \theta$
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