The local Gauss Bonnet theorem states that for any region $R$ in a regular surface $S$ that is homeomorphic to a closed disk and whose boundary is parametrized by a piecewise regular curve $\alpha$, then $$\sum_{i=1}^m \int_{J_i}k_g(s)ds + \int\int_R K d\sigma + \sum_{i=1}^m \theta_i = 2\pi$$ where $k_g$ is the geodesic curvature of the regular arcs of $\alpha$ and $K$ is the gaussian curvature of $S$.
I get that for a surface of constant $K$ this theorem can be used to easily compute the area enclosed by $\alpha$, though I can't see how to use it on a surface on which $K$ is not constant.
Take the hyperbolic hyperboloid $x^2 + y^2 - z^2 = 1$ parametrized by $$x(u,v) = (\cosh(v)\cos(u),\cosh(v)\sin(u),\sinh(v)).$$ The gaussian curvature of this is $K = -(\cosh(v)^2 + \sinh(v)^2)^{-1}$.
How can I compute the area in the region $0<v<w$ , $0<u<\phi$?