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 $\Delta$.

Also see

• Definition:Definite Integral
• Definition:Upper Sum
• Definition:Lower Sum

Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4.3$
• Weisstein, Eric W. "Riemann Sum." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannSum.html