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: 19


$2 \ln 3$

Work Step by Step

Here, we have: $|J| =\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix} \dfrac{1}{v}&-\dfrac{u}{v^2}\\0& 1\end{vmatrix}=v^{-1}$ $\iint_R xy dA=\int_1^{3} \int_{u^{1/2}}^{(3u)^{1/2}} [uv^{-1}] dv du=\int_1^3u[\ln v]_{u^{1/2}}^{(3u)^{1/2}} du$ and $\iint_R xy dA=\int_1^3u[\ln (3)^{1/2}]du=[\ln 3^{1/2}][\dfrac{u^2}{2}]_1^3$ Hence, $\iint_R xy dA=(\dfrac{1}{2}) (\ln 3) [\dfrac{9}{2}-\dfrac{1}{2}]=2 \ln (3)$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.