Evaluate $\iint_{D}e^\frac{x+y}{x-y}$ on the region $D$ using a change of variables

Question

Evaluate $\displaystyle\iint_{D}e^\dfrac{x+y}{x-y}$ on the region $D$ using a change of variables

Here is the region:

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The region is bounded by the following:

$y=x-2, y=x-1, x=0, y=0$.

A sub that is obvious is let $u=y-x$, then $-2\leq u\leq -1$.

Now I have to make one more sub that takes care of the $x=0, y=0$, so I made the sub, let $v=\frac{y}{y-x}$.

Then if $x=0\to v=1$, and if $y=0\to v=0$. Therefore $0\leq v\leq 1$.

Before I calculate the Jacobian, I notice that I cannot write $e^\frac{x+y}{x-y}$ in terms of $u,v$. The closest I can get is $e^\frac{x+y}{-u}$, but the $x+y$ is still left? Have I made an incorrect sub?

Answer 1

You said

$$u = y-x$$

Then go for this.

Since you wrote then

$$v = \frac{y}{y-x}$$

Then you can easily check that this means

$$v = \frac{y}{u}$$

hence

$$y = uv$$

From this you can eliminate $y$ completely, and going back you can also eliminate $x$, writing it in terms of $u, v$.

Answer 2

Just note that $$ \frac{x+y}{x-y}=\frac{x-y+2y}{x-y}=1+\frac{2y}{x-y}=1-2v. $$