Definition:Riemann Sum

Definition

Let $f$ be a real function defined on the closed interval $\mathbb I = \closedint a b$.

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

For $1 \le i \le n$:

let $\Delta x_i = x_i - x_{i - 1}$

let $c_i \in \closedint {x_{i - 1} } {x_i}$.

The summation:

$\ds \sum_{i \mathop = 1}^n \map f {c_i} \Delta x_i$

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

Geometric Interpretation

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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.): $\S 4.3$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Riemann integral, Riemann sum

• Weisstein, Eric W. "Riemann Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannSum.html