Evaluating a repeated integral by changing to polar co-ordinates

Okay, so I missed my lectures on this and I haven't a clue (Crying) I have my friend's assignment here but she's gone from step to step and I totally don't understand how. The question is:

Evaluate by changing to polar co-ordinates

$\displaystyle \int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}\sin(\frac{\pi(x^2+y^2)}{2})dydx$

My friend has then wrote down $\displaystyle x=r\cos\theta$ and $\displaystyle y=r\sin\theta$ then jumped to...

$\displaystyle \int_{0}^{\pi/2}\int_{0}^{1}\sin(\frac{\pi r^2}{2})r drd\theta$

...and I don't know how. Can anyone go through it explaining the steps to me?