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: 12


$x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$

Work Step by Step

Consider $u=3x-4y$ and $v=x+2y$ Given: The parallelogram with vertices $(0,0), (4,3), (2,4), (-2,1)$. which in the form of the equations as follows: $-10 \lt 3x-4y \lt 0$; $0 \lt x+2y \lt 10$ or, $-10 \lt u \lt 0$ and $0 \lt v \lt 10$ This represents a rectangle in the $uv$ plane. Now take the assumptions such as: Multiply $v=x+2y$ with $2$ and then take sum with $u=3x-4y$. we get $3x-4y+2x+4y=u+2v$ $\implies x=\dfrac{u+2v}{5}$ Also, $v=x+2y$ with $-3$ and take the sum with $u=3x-4y$. we get $3x-4y-3x-6y=u-3v$ This implies that $y=\dfrac{3v-u}{10}$ Hence, $x=\dfrac{u+2v}{5}$ and $y=\dfrac{3v-u}{10}$
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