Wow leave the forum for a few hours and there's 3 great replies waiting for me!!

Ahh yeh I've done Jacobian determinants before, it didn't click that it applied here.

I'd drawn the circle but for some reason only drew $\displaystyle y = x$ for positive values of $\displaystyle x$ and $\displaystyle y$, that would be where I was going wrong I guess

** Think you mean $\displaystyle \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos^2(\theta) \bigg( \int_0^{\sqrt{3}} r \, \, dr\bigg) \, d\theta $

Anyway on with the integral:

$\displaystyle \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos^2(\theta) \bigg( \int_0^{\sqrt{3}} r \, \, dr\bigg) \, d\theta $

$\displaystyle \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos^2(\theta) \bigg[ \frac{r^2}{2} \bigg]_0^{\sqrt{3}} \, d\theta $ = $\displaystyle \frac{3}{2}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \frac{1}{2}(1 + \cos{2\theta}) \, d\theta $ =

$\displaystyle \frac{3}{4}\bigg[1 + \cos{2\theta}\bigg]_{\frac{\pi}{4}}^{\frac{5\pi}{4}} $ = $\displaystyle \frac{3}{4}(\pi)$ = $\displaystyle \frac{3\pi}{4}$

Thanks a lot for the help!