Calc 3 question - change of variables and computing the jacobian?
The problem asks to use a change of variables compute the double integral of y4dA 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) = y2.
Similarly, dividing gives
u / v = (xy) / (y/x) = x2.
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.