When integrating a one dimensional (univariate) function $y=f(x)$ numerically from a to b, the naive way is to take steps from a to b, evaluate the function at each step, and sum the difference between each step.
When taking this to a two dimensional (bivariate) function $z=f(x,y)$, the naive way is to do this involves a rectangular area to sample.
A method used in software to make numeric Integration queries more efficient is a summed area table which precomputes a 2d table of an indefinite integral (sans the starting constants).
This extends to higher dimensions using hypercubes.
I was wondering, are there any methods for numeric Integration involving simplices? (Triangle, tetrahedron, etc)