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: 22


$2.5(b-a) \ln \dfrac{d}{c}$

Work Step by Step

$J(u,v)=\begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{vmatrix}$ or, $=\begin{vmatrix} 3.5u^{2.5}v^{-2.5}&-2.5v^{2.5}u^{-3.5}\\-2.5u^{3.5}u^{-3.5}& 2.5v^{1.5}u^{-2.5}\end{vmatrix}=\dfrac{2.5}{v}$ $\iint_R dA=\int_c^{d} \int_{a}^{b}\dfrac{2.5}{v} du dv=\int_c^d \dfrac{2.5}{v} dv \int_{a}^{b} du$ Therefore, $\iint_R dA=2.5[\ln d -\ln c](b-a)=2.5(b-a) \ln \dfrac{d}{c}$
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