As far as I understand this coonection connection was observed (and generalised) by Khovanskii and Puhlikov in the article
A. G. Khovanskii and A. V. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, Algebra and Analysis 4 (1992), 188–216, translation in St. Petersburg Math. J. (1993), no. 4, 789–812.
This is related to toric geometry, for which some really well written introduction articles are contained on the page of David Cox http://www3.amherst.edu/~dacox/
Since 1992 many people wrote on this subject, for example
EXACT EULER MACLAURIN FORMULAS FOR SIMPLE LATTICE POLYTOPES
Or Riemann sums over polytopes http://arxiv.org/PS_cache/math/pdf/0608/0608171v1.pdf