Using the Riemannian metric

$$d s^{2}=\frac{d x^{2}+d y^{2}}{y^{2}}$$

define the length of a curve and the area of a region in the upper half-plane $H=\{x+i y: y>0\}$.

Find the hyperbolic area of the region $\{(x, y) \in H: 0<x<1, y>1\}$.