This paper concerns with analytical integration of trivariate
polynomials over linear polyhedra in Euclidean three-dimensional space.
The volume integration of trivariate polynomials over linear polyhedra is
computed as sum of surface integrals in
on application of the well
known Gauss's divergence theorem
and by using triangulation of the linear polyhedral boundary. The
surface integrals in
over an arbitrary triangle are
connected to surface integrals of bivariate polynomials in
.
The surface integrals in
over a simple polygon or over an
arbitrary triangle are computed by two different approaches.
The first algorithm is obtained by transforming the surface
integrals in
into a sum of line integrals in a one-parameter
space, while the second algorithm is obtained by transforming
the surface integrals in
over an arbitrary triangle into a
parametric double integral over a unit triangle. It is shown that
the volume integration of trivariate polynomials
over linear polyhedra can be obtained as a sum of surface
integrals of bivariate polynomials in
.
The computation of
surface integrals is proposed in the beginning of this paper and
these are contained in Lemmas 1-6. These algorithms (Lemmas
1-6) and the theorem on volume integration are then followed by
an example for which the detailed computational scheme has been
explained. The symbolic integration formulas presented
in this paper may lead to an easy and systematic incorporation of
global properties of solid objects, for example, the volume,
centre of mass, moments of inertia
etc., required in
engineering design processes.
