Definition:Definite Integral/Riemann

Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function. Let $\Delta$ be a finite subdivision of $\left[{a \,.\,.\, b}\right]$, $\Delta = \left\{{x_0, \ldots, x_n}\right\}$, $x_0 = a$ and $x_n = b$.

Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \left({c_1, \ldots, c_n}\right)$, where $c_i \in \left[{x_{i - 1} \,.\,.\, x_i}\right]$ for every $i \in \left\{{1, \ldots, n}\right\}$.

Let $S \left({f; \Delta, C}\right)$ denote the Riemann sum of $f$ for the subdivision $\Delta$ and the sample point sequence $C$.

Then $f$ is said to be (properly) Riemann integrable on $\left[{a \,.\,.\, b}\right]$ :
 * $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivision $\Delta$ of $\left[{a \,.\,.\, b}\right]: \forall$ sample point sequences $C$ of $\Delta: \left\Vert{\Delta}\right\Vert < \delta \implies \left\vert{S \left({f; \Delta, C}\right) - L}\right\vert < \epsilon$

where $\left\Vert{\Delta}\right\Vert$ denotes the norm of $\Delta$.

The real number $L$ is called the Riemann integral of $f$ over $\left[{a \,.\,.\, b}\right]$ and is denoted:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$

More usually (and informally), we say:
 * $f$ is (Riemann) integrable over $\left[{a \,.\,.\, b}\right]$.

Also denoted as
The notation:
 * $\displaystyle \int_a^b f$

can occasionally be seen.

Also see

 * Equivalence of Definitions of Riemann and Darboux Integrals