Integration on three-dimensional regular regions based on (modified) euler summation

Most of the time three-dimensional volume integrals can not be determined analytically. Thus, approximate integration methods are necessary. We present an alternative to classical numerical integration for the case of three-dimensional regular regions-(modified) Euler summation formulas. Summarizing previous results of the authors, theoretical and numerical aspects of these methods are highlighted. The simplicity of the classical version of the Euler summation formula is based on the approximation of a volume integral by the weighted summation over all functional values of the integrand in lattice points in the closure of the regular region.