Now it seems that a substitution would be the best bet. What I've done is make u=xy, and v=x^2 -y^2. From this, I calculate the Jacobian and get |J|=2(x^2 +y^2). The problem is, I can't then find J in terms of u,v. I've tried many other substitutions, but finding the limits of integration just become much more difficult.

Can anyone either suggest another substitution (right or wrong doesn't matter, I'll check anyways) or give me a hint as to how to find J in terms of u,v? I've been doing this question for about 3-5 hours and I just keep going in circles.

Thank you very much