Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 1 / 3G / Solution

For a piecewise continuously differentiable curve $\gamma (t)=(x(t),y(t))$ in the upper half-plane model, the given Riemannian metric gives hyperbolic lengthThe metric matrix has determinant $y^{-4}$, so its area element is $y^{-2}dxdy$. Thus a measurable region $D$ has hyperbolic area ${\int }_D{y}^{-2}dxdy$. For the specified strip the integral isThe infinite Euclidean height is compatible with finite hyperbolic area because the area density decreases quadratically with height.