Find the area of a geodesic triangle in the hyperbolic space

Question

The hyperbolic space of dimension 2 can be modeled via the upper half plane $H^2 = \{(u, v) \in \mathbb{R^2} | v > 0\}$ equipped with the metric $g(u, v) = \frac{1}{v^2} I_2$, with $I_2$ the identity matrix of dimension 2. Let's consider the geodesic triangle in $H^2$ whose vertices are lying all at infinity. It consists of the outside of a half circle of radius 1 and enclosed inside two vertical lines which start at one of the ends of the diameter of the half circle and are drawn to infinity in the same direction. I need to calculate the area of that triangle by integration. I am fully confused with the shape of the triangle, the bounds of the integral and I do not know how to proceed with calculating it's area. Can you provide some suggestion or a solution proposal ? Thanks.