$[-/14\mathrm{.}5$ Points$]$

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Tutorial Exercise

Evaluate the given integral by making an appropriate change of variables. ${}_R\frac{x-2y}{3x-y}dA,$ where $R$ is the parallelopram enclosed by the lines $x-2y=0,x-2y=3,3x-y=1,$ and $3x-y=3$

Step $1$

Let $u=x-2y$ and $v=3x-y.$ Then in terms of $u$ and $v,x=\frac{2}{5}v-$ and $y=\frac{1}{5}y=$

Therefore,

$\frac{\text{d}\text{e}\text{l}(x,y)}{\text{d}\text{e}\text{l}(u,v)}=|[\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{v}}],[\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{v}}]|=|[-\frac{1}{5},\frac{2}{5}],[-\frac{3}{5},\frac{1}{5}]|=1$

$$ SKIP $($YOU CANNOT COME BACK$)$