I am trying to learn how to apply this formula to problems of finding the integral of the gaussian curvature:
$$\int_0^{l(\gamma)} \kappa_g(s) ds = \sum_{i=1}^n \alpha_i - (n-2)\pi - \int_{int(\gamma)} KdA, $$
where $\gamma$ is a curvilinear polygon, $l(\gamma)$ is the length of $\gamma$, and $\alpha_i$ are the inner angles at the corner points.
I have bunch of problems that I am trying to solve using this formula. For example, I'd like to compute $\int_{M_r} KdA$ where $$M_r = \{(x,y,z)\in \mathbb{R}^3 | x^2+y^2 = z < r^2; x,y>0 \}. $$
Finding the geodesic curvature and computing the integral of that should hopefully not be a problem for me to do on my own. Specifically, the issues I'm having have to do with finding a parameterization $\gamma$ that is a curvilinear polygon, and determining the inner angles at the corners of that polygon. I am studying on my own so I don't really have anywhere else to turn, and would be so grateful if someone wanted to guide me through this.