On the error of the approximate calculation of double integrals with variable upper limits

Abstract

In the paper, the estimations of the approximate calculation of double integrals with variable upper limits are suggested. The theorem on estimation for the main error term of approximate integration using the approach of substituting the integrand with an approximating function is formulated and proved, as well as the theorem on the estimation of the main error term in the case of calculating double integrals using the sequential integration approach. Estimations of integration errors are obtained when dividing the variable region of integration into rectangular grids, in particular, four possible options for adapting the grid to the change of the integration region are considered. For each of these options the estimations are found for rectangular and curvilinear elements of the grid as well as total estimation of double integration with variable upper limits.