We first consider the integration of arbitrary functions over a linear polygon with n-edges say Pn in the Cartesian 2-space. A scheme based on the use of classical Green’s theorem is proposed. In this scheme, the integration over Pn is obtained as a sum of n-one dimensional integrals over the oriented edges of Pn. Gauss Legendre quadrature rules are further applied to develop the numerical integration scheme. We next consider the polynomial spline interpolatios.The term spline refers to an instrument used in drafting. It is a thin, flexible wooden or plastic tool that is passed through given data points and defines a smooth curve in between. Spline interpolation is a very powerful and widely used method and has many applications in numerical differentiation and integration, solution of boundary value problems, computer animation, computer graphics and robot path/ trajectory planning. This study proposes a cubic spline which interpolates at first and the last knots and the two points located at the trisections between the knots. The performance of the new cubic spline is found to be superior to the existing cubic splines studied in the literature. We have also presented the applications of proposed cubic spline to integral function approximations and the numerical integration over curved domains in 2-space. Applications to integral function approximations are illustrated for the indefinite integral of Runge function, logarithmic function and normal distribution. This application is very useful in the construction of table of integrals. As an application of numerical integration over curved domains the computation of some typical test integrals over a lunar model is considered. We also compare the numerical approximations of the proposed new cubic spline with the standard cubic splines (natural, clamped and not a knot)