Riemann sum

Named after Bernhard Riemann, a German mathematician.

Riemann sum (plural Riemann sums)

1. (mathematics) A certain kind of approximation of an integral by a finite sum that is calculated by dividing the interval of integration into smaller sub-intervals and summing sample values of the integrand inside those sub-intervals multiplied by their lengths.

In mathematical terms, let ${\displaystyle f}$ be a function defined on a closed interval ${\displaystyle [a,b]}$. Let ${\displaystyle [a,b]}$ be partitioned by points ${\displaystyle a=x_{0}<x_{1}<x_{2}<\cdots <x_{n-1}<x_{n}=b}$, and ${\displaystyle \Delta x_{k}}$ denote the length of the ${\displaystyle k}$th subinterval ${\displaystyle [x_{k-1},x_{k}]}$, i.e. ${\displaystyle \Delta x_{k}=x_{k}-x_{k-1}}$. A Riemann sum ${\displaystyle S}$ of ${\displaystyle f}$ is defined as ${\displaystyle S=\sum _{k=1}^{n}f(x_{k}^{*})\,\Delta x_{k}}$, where ${\displaystyle x_{k}^{*}}$ is an arbitrary point in the ${\displaystyle k}$th subinterval, i.e. ${\displaystyle x_{k}^{*}\in [x_{k-1},x_{k}]}$.

• Riemann integral