Definition:Riemann Sum
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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:
- $\displaystyle \sum_{i \mathop = 1}^n \map f {c_i} \Delta x_i$
is called a Riemann sum of $f$ for the subdivision $\Delta$.
Geometric Interpretation
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Also see
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. http://mathworld.wolfram.com/RiemannSum.html
