Double integral transformation

[vampireyal]

Homework Statement

evaluate the integral \int\int(x^4-y^4)e^{xy}dA

where R is the region bounded by xy=1, xy=2, x²-y²=1, and x²-y²=4

Homework Equations

The Attempt at a Solution

This is my first time on the forum, so forgive me if there are mistakes in this post. I am trying to find the transformation equations in order to convert the xy-plane equations to a uv-plane. Judging by the boundary equations, I let u=xy and v=x²-y².

When I evaluate the xy equations using the transformation equations I get u=1, v=4, u=2, and v=1. which makes a square on the uv plane. I am not sure if I did this correctly though.

The next part of the problem asks to calculate the Jacobian, but I am not sure of how to calculate the partial derivatives based off my transformation equations. I can't seem to get x and y in terms of u and v, which makes me think that my transformation equations are wrong.

Any help is appreciated. Thank you.

 

[Fragment]

What is 'SUP' supposed to stand for exactly? I'm assuming it's a mistake when you typed your formula...

 

[vampireyal]

oh...that's funny that did that...it's just supposed to be superscripted...so int int (x^4-y^4)e^xy dA

 

[HallsofIvy]

Don't use "sup" inside LaTex. Use "^" instead.