Ok, I've seen some questions similar to mine but it didn't really get me what I want so I figured I'd ask. I was given the following problem to solve by making the change of variables $ u = x-y, v = x+y$ in the following integral
$ I = \int_0^1dy\int_0^{1-y}e^\frac{x-y}{x+y}dx$
This is an iterated integral over an region of the x-y plane. The Jacobian of the transformation is 1/2 so the integral becomes
$I = \frac{1}{2}\int\int_Ae^\frac{u}{v}dudv $
At this point I had some difficulties. I eventually realized that if I played with the numbers, that u would have upper and lower bounds of 1 and -1 while v would have upper and lower bounds of 1 and 0. So my first attempt was this:
$ I = \frac{1}{2} \int_0^1dv\int_{-1}^1e^\frac{u}{v}du$
however evaluating this integral was practically impossible (if you don't believe me, try finding an antiderivitive for $xsinh(1/x)$ with elementary techniques!), eventually I realized the integral would simplify if changed the limits as follows:
$ I = \frac{1}{2} \int_0^1dv\int_{-v}^ve^\frac{u}{v}du$
this yielded the correct answer [$I=0.5sinh(1)$] however I would like to know how to determine how to find regions of integration on change of variables in ways other than guessing at the answer! Is there an algorithmic procedure? Is there an easy graphical procedure? I know I could graph in the u-v plane but that seems like a lot more work than necessary.
edit: the way I am thinking about this now is that the first set of boundaries are incorrect because they correspond to a region that is a box (rectangle) in the uv plane. Cleary, the region is triangular in the x-y plane, and I'm assuming it is also triangular in the uv plane. How do I determine the dependence between u and v in of this region?