change of variables formula for integrals

Does anyone have any hints on how to figure out a change of variables formula for integrals? There aren't many tips in the book, and I just can't figure this one out. I think the fact that $\displaystyle f(x,y)$ isn't linear is what's messing me up.

Here is the problem:

$\displaystyle \int\int_R\sin(9x^2+4y^2)dA$ where $\displaystyle R$ is the region in the first quadrant bounded by the ellipse $\displaystyle 9x^2+4y^2=1$.

I guess I could do $\displaystyle u=9x^2+4y^2$, but what about $\displaystyle v$? What am I missing? Thank you!