Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

Abstract

This paper is concerned with explicit formulas and algorithms for computing integrals of polynomials over a linear polyhedron in Euclidean three-dimensional space. Symbolic formulas for surface and volume integration are given. Two different approaches are discussed: The first algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric line integral while the second algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric double integral. These algorithms and formulas are followed by an application-example for which we have explained the detailed computational scheme. The symbolic results presented in this paper may lead to an easy incorporation of global geometric properties of solid objects, for example, the volume, centre of mass, moments of inertia, required in the engineering design process.