Double integral coordinate transform

Basically I want to find the new limits w,x,y,z when we make the valid transformation

I've tried putting in arbitrary functions f, and so getting 4 equations constraining the limits, but I end up with complicated equations involving exponentials, which I don't think I'd be able to solve. Also because of the original singular upper limits, the functions have to be of exponential form like e^(-t_1) e^(-t_2) etc... which means that I'm finding it difficult to come up with a fourth constraint which is different from the other 3 I have found...

Basically if I make a coordinate transform as above, considering a double integral as above, how do I find the new limits?