Multiple integral change of variables

[BrownianMan]

Let R denote the region inside x^2 + y^2 = 1 but outside x^2 + y^2 = 2y with x=>0 and y=>0. Let u=x^2 + y^2 and v=x^2 + y^2 -2y. Compute the integral of x*e^y over the region D in the uv-plane which corresponds to R under the specified change of coordinates.

I'm having trouble with this one. My first attempt at figuring out the new limits of integration yielded 0<=u<=1 and 0<=v<=u, which seems wrong to me. I'm also not sure how to change the integrand to make it a function of u and v.

 

[spamiam]

I'm still trying to work out the new limits of integration myself, but I think it might help to complete the square for the second condition...

As for changing the function's variables, can you solve for y in terms of u and v? (Hint: What is u-v?)