Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 14 - Multiple Integrals - 14.3 Double Integrals In Polar Coordinates - Exercises Set 14.3 - Page 1024: 15

Answer

\[ 2 \int_{0}^{\frac{\pi}{2}} \int_{0}^{\cos \theta}\left(-r^{2}+1\right) r \operatorname{drd} \theta \]

Work Step by Step

\[ \begin{array}{c} 1-x^{2}-y^{2}=z \\ 1-r^{2}=z \\ 0=x^{2}+y^{2}-x \\ r^{2}-r \cos \theta=0 \\ \cos \theta=r \text { or, } 0=r \end{array} \] The integral is: \[ 2 \int_{0}^{\frac{\pi}{2}} \int_{0}^{\cos \theta}\left(-r^{2}+1\right) r \operatorname{drd} \theta \]
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