Calculus 8th Edition

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

Chapter 15 - Multiple Integrals - 15.9 Change of Variables in Multiple Integrals - 15.9 Exercises - Page 1100: 13


$x =u \cos v$ and $y =u \sin v$ where $S=${$(u,v) | 1 \leq u \leq \sqrt 2, 0\leq v \leq \dfrac{\pi}{2}$}

Work Step by Step

Here, we have $u=\sqrt{x^2+y^2}$ and $v=\tan^{-1} \dfrac{y}{x}$ and $ y=x \tan v$ Therefore, we get $1 \leq u \leq \sqrt 2, 0\leq v \leq \dfrac{\pi}{2}$ Further, $u=\sqrt{x^2+x^2 \tan^2 v}=x\sqrt {1+\tan^2 v}$ This implies that $u=x \sec v=\dfrac{x}{\cos v}$ or, $x =u \cos v$ and $\tan v=\dfrac{y}{u \cos v} \implies y=u \sin v$ Hence, $x =u \cos v$ and $y =u \sin v$ where $S=${$(u,v) | 1 \leq u \leq \sqrt 2, 0\leq v \leq \dfrac{\pi}{2}$}
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