Definition:Definite Integral/Riemann

Definition
Let $\closedint a b$ be a closed real interval.

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

Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \tuple {c_1, \ldots, c_n}$, where $c_i \in \closedint {x_{i - 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$.

Let $\map S {f; \Delta, C}$ 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 $\closedint a b$ :
 * $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivisions $\Delta$ of $\closedint a b: \forall$ sample point sequences $C$ of $\Delta: \norm \Delta < \delta \implies \size {\map S {f; \Delta, C} - L} < \epsilon$

where $\norm \Delta$ denotes the norm of $\Delta$.

The real number $L$ is called the Riemann integral of $f$ over $\closedint a b$ and is denoted:
 * $\displaystyle \int_a^b \map f x \rd x$

More usually (and informally), we say:
 * $f$ is (Riemann) integrable over $\closedint a b$.

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

can occasionally be seen.

Also see

 * Equivalence of Definitions of Riemann and Darboux Integrals