# Symmetry and the Monotonicity of Certain Riemann Sums

Conference paper

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## Abstract

We consider conditions ensuring the monotonicity of right and left Riemann sums of a function \(f:[0,1]\rightarrow \mathbb {R}\) with respect to uniform partitions. Experimentation suggests that symmetrization may be important and leads us to results such as: *if* *f* *is decreasing on* [0, 1] *and its symmetrization,* \(F(x) := \frac{1}{2}\left( f(x) + f(1-x)\right) \) *, is concave then its right Riemann sums increase monotonically with partition size.* Applying our results to functions such as \(f(x) = 1/\left( 1+x^2\right) \) also leads to a nice application of Descartes’ rule of signs.

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