Optimizing Limits of Integration for Change of Variables

[eyesontheball1]

Hi guys, I've been on quite a random change of variables binge lately and I've been messing around with a particular scenario in which I'm not 100% sure of how I should choose my limits of integration. Any help would be greatly appreciated! (And no, this is not homework, etc.) The scenario is as follows:

Domain of integration in the xy plane is some rectangle in the 1st quadrant with vertices (a,c), (b,c), (a,d), (b,d), and I want to make a change of variables using x = u + $\sqrt{u^2 - v^2}$ and y = u - $\sqrt{u^2 - v^2}$. Other than the fact that this transformation would inherently require v≤u, I'm just not certain on how to determine the other numerical limits of integration for u and v.

 

[eyesontheball1]

Update: I think I may have the bounds correct now. If not, someone please correct me. To make this change of variables, you have to express the new integration problem as three separate double integrals, with each of the double integrals having the following general limits of integration:

DI#1: $\frac{a+c}{2}$≤u≤$\sqrt{bd}$ ; $\frac{a+c}{2}$≤v≤u

DI#2: $\frac{a+c}{2}$≤u≤$\frac{b+d}{2}$ ; $\sqrt{ac}$≤v≤$\frac{a+c}{2}$

DI#3: $\sqrt{bd}$≤u≤$\frac{b+d}{2}$ ; $\frac{a+c}{2}$≤v≤$\sqrt{bd}$