Guessab, Noouisser, and Schmeisser "A Definiteness Theory for Cubature Formulae of Order Two", Constructive Approximation (2006)24:263-288
Define a quantity $R[||\cdot||^2]$ which is $$\sum_{i=1}^N \int_{\Omega_i} ||\underline{x}-\underline{x_i}||^2\,dx$$ where $\Omega_i$ is the Voronoi cell of node/sample-point $\underline{x_i}$ (and there are N points).
I would like to calculate $R[||\cdot||^2]$ for a sample design embedded on the surface of a sphere. Calculating the Voronoi vertices and from them the area of the fractional spherical triangles making up the Voronoi cell is complicated but already solved. Calculating
But I'm having trouble doing $$\int_{\omega_{i,j}} ||\underline{x}-\underline{x_i}||^2dx$$ where $\Omega_i=\bigcup_{i=1}^t\omega_{i,j}$ i.e. the $\omega_{i,j}$ is one of the $t$ spherical triangle components of the Voronoi cell i, which has one corner at node $\underline{x_i}$ and 2 corners that are adjacent voronoi vertexes.
If someone could provide the solution to $$\frac{1}{4}\int_0^{\alpha}\left(\cos^{-1}\left(Y\sin(\theta)+Z\cos(\theta)\right)\right)^3d\theta$$ it would solve the larger problem for me (this is a simpler and hopefully easier to integrate formulation of the same problem it replaced), everything in the last equation except $\theta$ is a constant