  Recently there has been a renewed interest in asymptotic Euler-MacLaurin
formulas, partly due to applications to spectral theory of differential
operators. Using elementary means, we recover such formulas for compactly
supported smooth functions f on intervals, polygons, and 3-dimensional
polytopes, where the coefficients in the asymptotic expansion are sums of
differential operators involving only derivatives of f in directions normal to
the faces of the polytope. Our formulas apply to wedges of any dimension. This
paper builds on, and is motivated by, works of Guillemin, Sternberg, and
others, in the past ten years.
