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 $S \left({ f; \Delta }\right)$ denote the Riemann sum of $f$ for a subdivision $\Delta$ of $\left[{a \,.\,.\, b}\right]$.

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

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 often denoted as:
 * $\displaystyle \int_a^b f$

or:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$

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

Also see

 * Equivalence of Definitions of Riemann Integral