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: 27



Work Step by Step

$Jacobian, J(x,y)=\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}=(1) \cdot (-1) -(1)(1)=-2$ $J(u,v)=|-\dfrac{1}{2}|=\dfrac{1}{2}$ Let us consider that $u=x+y$ and $v=x-y$ Now, $\iint_R e^{x+y} dA= \int_{-1}^{1} \int_{-1}^{1} (\dfrac{1}{2})(e^u) du dv$ or, $\iint_R e^{x+y} dA= \int_{-1}^1 (\dfrac{1}{2}) |e^u|_{-1}^1 dv= \dfrac{1}{2}(e-e^{-`1}) \int_{-1}^1 dv$ Hence, our results is: $\iint_R e^{x+y} dA=e-e^{-1}$
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