# Double integral change of variables

Using an appropriate change of variables, evaluate $$\iint_{B}\exp\left(\,{y - x \over y+x}\,\right)\,{\rm d}x\,{\rm d}y$$ Where $B$ is the interior of the triangle with vertices at $\left(\, 0,0\,\right), \left(\, 0,1\,\right)\ \mbox{and}\ \left(\, 1,0\,\right)$.

Attempt:

I used $u = y - x\,,\ v = y + x$ as my new variables. I also found that the old bounds were $0\ \leq\ x\ \leq\ 1$ and $0\ \leq\ y\ \leq\ 1 - x$.

However, I don't understand how to put my bounds in terms of $u$ and $v$.

I tried setting $y = u + x = v - x$, then solving for $x$ to get $x = \left(\, v - u\,\right)/2$. I substituted that into the bound for $x$ and got $0\ \leq\ v\ \leq\ 2 + u$.

Is that correct ?. How do I get the constant bounds for $u$ ?.

Thank you!

• I suspect a typo, as as stated your integrand is $e^0$ – GFauxPas Nov 12 '14 at 22:32
• You're right! I fixed it. – sosoo Nov 12 '14 at 22:57

The points $(0,0), (1,0), (0,1)$ in the xy-plane map to the points $(0,0), (-1,1), (1,1)$ in the uv-plane; so
the region should be described by the inequalities $0\le v\le 1$, $-v\le u\le v$.