The paper proposes an efficient and accurate method for integrating the basins of attraction of a smooth function defined on a discrete grid, specifically applied to Bader charge partitioning for electron charge density. The method involves deriving an expression for the fraction of space neighboring each grid point that flows to its neighbors, which is used to compute the fraction of each grid volume that belongs to a basin (Bader volume) and as a weight for the discrete integration of functions over these volumes. Compared to other grid-based algorithms, this approach is robust, computationally efficient with linear computational effort, accurate, and has quadratic convergence. It is also straightforward to extend to non-uniform grids and can be used to identify basins of attraction of fixed points and integrate functions over these basins. The method is demonstrated through various examples, including three-dimensional charge density from Gaussian functions in FCC cells, TiO$_2$ bulk, and NaCl crystal, showing improved computational efficiency and accuracy.The paper proposes an efficient and accurate method for integrating the basins of attraction of a smooth function defined on a discrete grid, specifically applied to Bader charge partitioning for electron charge density. The method involves deriving an expression for the fraction of space neighboring each grid point that flows to its neighbors, which is used to compute the fraction of each grid volume that belongs to a basin (Bader volume) and as a weight for the discrete integration of functions over these volumes. Compared to other grid-based algorithms, this approach is robust, computationally efficient with linear computational effort, accurate, and has quadratic convergence. It is also straightforward to extend to non-uniform grids and can be used to identify basins of attraction of fixed points and integrate functions over these basins. The method is demonstrated through various examples, including three-dimensional charge density from Gaussian functions in FCC cells, TiO$_2$ bulk, and NaCl crystal, showing improved computational efficiency and accuracy.