The classic idea to relate the maximum of a function over a discrete or continuous domain to certain sums or integrals has made its apppearance in a number of recent papers from the point of view of optimization (A.I. Barvinok, "Exponential integrals and sums over convex polyhedra", Funktsional. Anal. i Prilozhen. 26 (1992); J.B. Lasserre, "Generating functions and duality for integer programs", Discrete Optim. 1 (2004)). Efficient summation and integration procedures can give rise to efficient approximation algorithms for optimization problems. As an example, a fully polynomial-time approximation scheme for optimizing arbitrary polynomial functions over the integer points in polyhedra of arbitrary, fixed dimension has been obtained (work with J. A. De Loera, R. Hemmecke, R. Weismantel, Math. Oper. Res. 31 (2006)). In this talk I report on recent work (with V. Baldoni, N. Berline, J. A. De Loera, M. Vergne) to study the efficiency of integration procedures for polynomial functions in high dimension. The methods are related to Brion's formulas, Barvinok's exponential sums, and also to the polynomial Waring problem that asks to represent a polynomial as a sum of powers of few linear forms.