Riemann sum

English

Etymology

Named after Bernhard Riemann, a German mathematician.

Noun

1. Template:Lb 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 $\mathrm{\Delta }{x}_k$ denote the length of the $k$th subinterval $[x_{k-1},x_k]$, i.e. $\mathrm{\Delta }{x}_k=x_k-x_{k-1}$. A Riemann sum $S$ of $f$ is defined as $S=\sum _{k=1}^{n}f(x_k^{*})\mathrm{\Delta }{x}_k$, where $x_k^{*}$ is an arbitrary point in the $k$th subinterval, i.e. $x_k^{*}\in [x_{k-1},x_k]$.

Translations

• Polish: Template:T

See also

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