Define the hyperbolic metric in the upper half-plane model $H$ of the hyperbolic plane. How does one define the hyperbolic area of a region in $H$ ? State the Gauss-Bonnet theorem for hyperbolic triangles.

Let $R$ be the region in $H$ defined by

$$0<x<\frac{1}{2}, \quad \sqrt{1-x^{2}}<y<1$$

Calculate the hyperbolic area of $R$.