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 $\Delta$.

Also see

 * Definition:Definite Integral
 * Definition:Upper Sum
 * Definition:Lower Sum