Double integral coordinate transform

[MadMax]

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

\int^{\infty}_0 \int^{\infty}_0 f(t_1,t_2) dt_1 dt_2 = \int^w_x \int^y_z f(st, s(1-t)) s dt ds

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?

 

[lalbatros]

You need to visualise the mapping from coordinate (s,t) to (t1,t2) (or reverse)
with t1=st and t2=s(1-t) (or s=t1+t2 and t=t1/(t1+t2)).
In this mapping you have to check how the boundaries are transformed,
and to check if there is no tear in the domain.
In case there is a tear curve, you need to divide your domain along the tear.