Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 15 - Section 15.9 - Change of Variables in Multiple Integrals - 15.9 Exercise - Page 1060: 20



Work Step by Step

Here, we have: $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} 2u/v&-v/u^2\\\dfrac{-u^2}{v^2}& \dfrac{1}{u}\end{vmatrix}=\dfrac{1}{v}$ $\iint_R y^2 dA=\int_1^{2} \int_{1}^{2}(\dfrac{v}{u})^2(\dfrac{1}{v}) du dv$ or, $=\int_1^2 v dv \int_{1}^{2} \dfrac{1}{u^2} du$ or, $=[v^2/2]_1^2[\dfrac{-1}{u}]_1^2$ Thus, we have $\iint_R y^2 dA=\dfrac{3}{4}$
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