Calc 3 question - change of variables and computing the jacobian?

The problem asks to use a change of variables compute the double integral of y^{4dA} over R, where R is the region bounded by the hyperbolas xy=1 and xy=4 and the lines y/x=1 and y/x=3.

I was able to find that u=xy and v=y/x, and that u goes from 1 to 4 and v goes from 1 to 3. However, I can't compute the Jacobian determinant for the life of me! When I try to solve for x and y, I get expressions that have both u and v and x and y; I can't seem to isolate x and y. Please help me! Thanks.

Re: Calc 3 question - change of variables and computing the jacobian?

Looks like all you need is x = x(u,v), y = y(u,v).

Well try multiplying your two expressions together. That is,

u * v = (xy) * (y/x) = y^{2}.

Similarly, dividing gives

u / v = (xy) / (y/x) = x^{2}.

Now you have,

x = (u/v)^{1/2}, y = (u*v)^{1/2}.

Hope that's enough to get you on the right track. Let me know otherwise and I'll try and clear it up.