Definition:Integral (Calculus)
Definition
Indefinite Integral
Let $F$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let $f$ be a real function which is continuous on the open interval $\openint a b$.
Let:
- $\forall x \in \openint a b: \map {F'} x = \map f x$
where $F'$ denotes the derivative of $F$ with respect to $x$.
Then $F$ is a primitive of $f$, and is denoted:
- $\ds F = \int \map f x \rd x$
Definite Integral
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 $\closedint a b$ if and only if:
- $\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ 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:
- $\ds \int_a^b \map f x \rd x$
Also see
- Results about integral calculus can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integral: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integral: 1.