= 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 length>
$$
\boxed{L(\gamma)=\int\frac{\sqrt{\dot x(t)^2+\dot y(t)^2}}{y(t)}\,dt.}
$$
The metric matrix has determinant $y^{-4}$, so its area element is $y^{-2}\,dx\,dy$. Thus a measurable region $D$ has <hyperbolic area> $\int_Dy^{-2}\,dx\,dy$. For the specified strip the integral is
$$
\boxed{\int_0^1\int_1^\infty\frac{dy\,dx}{y^2}=1.}
$$
The infinite Euclidean height is compatible with finite <hyperbolic area> because the area density decreases quadratically with height.
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