# Definition:Riemann Sum

## Definition

Let $f$ be a real function defined on the closed interval $\mathbb I = \left[{a \,.\,.\, b}\right]$.

Let $\Delta$ be a subdivision of $\mathbb I$.

For $1 \le i \le n$:

let $\Delta x_i = x_i - x_{i - 1}$
let $c_i \in \left[{x_{i - 1} \,.\,.\, x_i}\right]$.

The summation:

$\displaystyle \sum_{i \mathop = 1}^n f \left({c_i}\right) \Delta x_i$

is called a Riemann sum of $f$ for the subdivision $\Delta$.

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.