Calculus 8th Edition

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

Chapter 15 - Multiple Integrals - 15.7 Triple-Integrals in Cylindrical Coordinates - 15.7 Exercises - Page 1083: 3


a) $(\sqrt 2, \dfrac{3\pi}{4}, 1)$ b) $(4, \dfrac{2\pi}{3}, 3)$

Work Step by Step

In the cylindrical coordinate system, we have $x=r \cos \theta \\ y=r \sin \theta \\z=z$ Conversion of rectangular to cylindrical coordinate system, we have $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$ a) $r=\sqrt{x^2+y^2} \implies r=\sqrt 2$ $\tan \theta=\dfrac{y}{x}$ $\implies \theta=\arctan (\dfrac{-1}{2})=\dfrac{3\pi}{4}$ Hence, $(r,\theta,z)=(\sqrt 2, \dfrac{3\pi}{4}, 1)$ b) $ r=\sqrt{(-2)^2+(2\sqrt 3)^2}=\sqrt{4+12}=4$ $x=r \cos \theta \implies -2=4 \cos \theta$ This gives: $\theta=\dfrac{2\pi}{3}$ Hence, $(r,\theta,z)=(4, \dfrac{2\pi}{3}, 3)$
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