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: 10

Answer

a) $z^2=2r^2-4$ and b) $z=1-2 r\cos \theta+r \sin \theta$

Work Step by Step

a) 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 $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$ $2(x^2+y^2)-z^2=4$ Re-arrange it as: $2r^2-z^2=4$ or, $z^2=2r^2-4$ b) 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 $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$ Now, $2r\cos \theta-r\sin \theta+z=1$ Re-arrange the above equations as: $z=1-2 r\cos \theta+r \sin \theta$
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