Minimizing numerical error in computing integrals over polygonal domains

Abstract. Integration of multivariable polynomials over polygons and polyhedra is an intensively studied topic of vector calculus. Physical properties of \(n\)-dimensional solids as volume, centroid and moment of inertia can be defined by this type of integrals. In many physical applications, it is essential to be able to quickly calculate integrals of polynomials. In this paper, we show a technique to calculate integrals of continuous functions over two-dimensional polygonal domains. Particularly, our results contain the known formulas for monomials and polynomials, and we will show how we can generate formulas that can be used to calculate integrals of arbitrary continuous functions. It becomes clear that a significant numerical error can occur, that depends on the direction of the edges of the polygon. We will show that a rotation by an angle \(\theta\) can solve this problem. Moreover we will give the optimal rotation angle \(\theta\), where the numerical error acquired from the orientation of the polygon is minimal.