Hi, 

In a recent post  (Monte Carlo Integration) Radaar shared its work about the numerical integration, with the Monte Carlo method, of a function defined in polar coordinates.
Radaar used a raw strategy based on a sampling in cartesian coordinates plus an ad hoc transformation.
Radaar obtained reasonably good results, but I posted a comment to show how Monte Carlo summation in polar coordinates can be done in a much simpler way. Behind this is the choice of a "good" sampling distribution which makes the integration problem as simple as Monte Carlo integration over a 2D rectangle with sides parallel to the co-ordinate axis.

This comment I sent pushed me to share the present work on Monte Carlo integration over simple polygons ("simple" means that two sides do not intersect).
Here again one can use raw Monte Carlo integration on the rectangle this polygon is inscribed in. But as in Radaar's post, a specific sampling distribution can be used that makes the summation method more elegant.

This work relies on three main ingredients:

1. The Dirichlet distribution, whose one form enables sampling the 2D simplex in a uniform way.
2. The construction of a 1-to-1 mapping from this simplex into any non degenerated triangle (a mapping whose jacobian is a constant equal to the ratio of the areas of the two triangles).
3. A tesselation into triangles of the polygon to integrate over.

This work has been carried out in Maple 2015, which required the development of a module to do the tesselation. Maybe more recent Maple's versions contain internal procedures to do that.
 

Monte_Carlo_Integration.mw