Finding $\iint _De^{\frac{x-y}{x+y}} dy\,dx$ where $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$

Question

Integrate $e^{\frac{x-y}{x+y}} dy\,dx$ over the set $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$.
I tried to do substitution $u=x-y$ and $v=x+y$ so i know that $ 1\leq v\leq 2$ but i couldn't figure how to get the boundaries for u

Answer 1

To find bounds on $u$, simply convert the given inequalities in $D$: $x \geq 0$ becomes $2x \geq 0$ which can be written as $u+v = 0$, and similarly, $y \geq 0$ can be written as $v - u \geq 0$. $1 \leq x+y \leq 2$ becomes $1 \leq v \leq 2$, as you've noticed.

Do be careful; changing variables to $u$ and $v$ requires you to multiply a Jacobian factor when you integrate.