Math Help - Gauss Bonnet theorem for hyperbolic triangle

1. Gauss Bonnet theorem for hyperbolic triangle

Hi, i was hoping that someone could help me understand the integration parts of the proof on page 3 of: http://www.maths.manchester.ac.uk/~c.../lecture07.pdf
Particularly i dont understand where the sqrt(1-x^2) comes from on the integral and the beta and pi - alpha

2. the half circle is the graph of $\sqrt{1-x^2}$. To compute
$\int_{\Delta}{\frac{1}{y^2}dxdy}$, note the area $\Delta$ is bounded by x=a, x=b and the half circle. So we can compute the integral using iterated integrals by Fubini's theorem, for x from a to b, y from $\sqrt{1-x^2}$ to infinite.