Double integral over region surrounded by two ellipses

pandarean

1. The problem statement, all variables and given/known data

A thin plate has the form of the intersection of the regions limited by [itex]\frac{x^2}{9}[/itex] + [itex]\frac{y^2}{4}[/itex] = 1 and [itex]\frac{x^2}{4}[/itex] + [itex]\frac{y^2}{9}[/itex] = 1

Which is the plate's mass if his density is δ(x, y) = |x|


2. The attempt at a solution

I've tried using u, v substitution

u = [itex]\frac{x^2}{4}[/itex] + [itex]\frac{y^2}{9}[/itex]

v = [itex]\frac{x^2}{9}[/itex] + [itex]\frac{y^2}{4}[/itex]

The resulting region looks nice, but the Jacobian is the ugly thing... i'm stuck.

I don't think polar is the way to go, one ellipse becomes a nice circle, but the other one becomes another ellipse....

Can someone give me some advice?
Thanks