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