[SOLVED] Green's Theorem and converting to polar co-ords

The question goes...

*Use Green's Theorem to evaluate $\displaystyle \oint_{\Gamma}(y^3dx-x^3dy)$, where $\displaystyle \Gamma$, mapped counter-clockwise, is the circle $\displaystyle x^2+y^2=4$. [Hint: you may find it convenient to introduce the transformation to polars.]*

Letting $\displaystyle x=\cos{\theta}$ and $\displaystyle y=\sin{\theta}$ and differentiating et cetera, I end up with...

$\displaystyle -\int_{0}^{2\pi}\sin^{4}{\theta}+\cos^{4}{\theta} d\theta$

And I'm stuck from here. I'm guessing it's something to do with relationship $\displaystyle \cos^{2}{\theta}+\sin^{2}{\theta}=1$ somewhere, but I'm just not getting it. Can anyone help?