This paper presents a numerical integration formula for the evaluation of ( )


∬ ( )


where ( ) and is any polygonal domain in . That is a closed domain with


boundary composed of N oriented piecewise line segments ( ) with end points


( ), ( ) and ( ) (( ). We join each of these line segments to a reference point


( ) interior to the domain . This creates a coarse mesh of N triangles ( ) in and


each of these arbitrary triangles have three straight sides. These arbitrary triangles can be divided into


arbitrary triangles ( ,2,3,…. ). by using the triangular mesh generation scheme


developed in this paper. We transform each = (say) into a standard 1-square and then into a 2-square


which can be integrated by using Gauss Legendre quadraure rules.We obtain enhanced accuracy by


division of these arbitrary triangles into four arbitrary triangles (e=1,2,3,4) without refining the already


generated triangular mesh.We first derive three different integration formulas for the integral


∬ ( )


where (( ) ) Then we


establish the necessary numerical integration formulas which use the well known Gauss Legendre


quadrature rules. Proposed numerical integration formula is applied to integrals over triangular and


polygonal domains with complicated integrands.