Riemann sum

English

Etymology

Named after Bernhard Riemann, a German mathematician.

Noun

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.

Usage notes

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

See also

• Riemann integral