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A simple and systematic method is described for calculating the integral of a polynomial function over an arbitrary nonconvex polyhedron. First a general formula is presented for direct evaluation of the integral of a polynomial over a 3-D simplex. An integral over a polyhedron can then be easily calculated by using the central projection method and decomposing a polyhedron symmetrically into a set of simplices and accumulating the results from each simplex based on this formula. This method adopts a systematic and automatic decomposition. It is analytically exact, but the practical accuracy of the result is within the accuracy of floating-point arithmetic. Furthermore, the time complexity of this method is linearly proportional to the number of vertices of a polyhedron.
Lien et al. (Mon,) studied this question.