Integration of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space

Abstract

In this paper, we present explicit integration formulas and algorithms for computing integrals of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space. Two different approaches are discussed: the first algorithm/formula is obtained by mapping the arbitrary tetrahedron into a unit orthogonal tetrahedron, while the second algorithm/formula computes the required integral as a sum of four integrals over the unit triangle. These algorithms/formulas are followed by an example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with the previous work. Further, it is shown that the present algorithms are much simpler and more economical as well in terms of arithmetic operations.