I find it confusing when there's a change of variable involved in a definite integral (double or triple) - in cases where the limits themselves are variable.
Here's an example to highlight my problem:
$$\int_0^1\int_0^{1-x}e^{\frac{x-y}{x+y}}~dx~dy$$
The change of variable to go for is $x-y=u$ and $x+y=v$
but now, what shall be the limits of the new integral?
A friend says that the same region in the $uv$ plane would be bounded by $u=-v$,$u=v$ and $v=1$. Could someone please explain exactly how this works? Any help is appreciated, thank you!
Refer to the diagram for the conversion below,
$$\int_0^1dx\int_0^{1-x}e^{\frac{x-y}{x+y}}dy=\frac12\int_0^1 dv \int_{-v}^{v}e^{\frac uv}du$$