User:Keith.U/Definite Integral
Definition: Definite Integral
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a real function.
Riemann Integrable
Definition 1
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 $\left[{a \,.\,.\, b}\right]$ if and only if there exists some $L \in \R$ such that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ 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 $\Delta$.
Definition 2
Let $f$ be bounded on $\closedint a b$.
Suppose that:
- $\ds \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$
where $\ds \underline {\int_a^b}$ and $\ds \overline {\int_a^b}$ denote the lower Darboux integral and upper Darboux integral, respectively.
Then $f$ is said to be (properly) Riemann integrable on $\closedint a b$.
More usually (and informally), we say:
- $f$ is (Riemann) integrable over $\closedint a b$.
Definite Integral
Riemann Integral
The real number $L$ as defined above is called the Riemann integral of $f$ over $\closedint a b$ and is denoted:
- $\ds \int_a^b \map f x \rd x$
Darboux Integral
The Darboux integral of $f$ over $\closedint a b$ is denoted
- $\ds \int_a^b \map f x \rd x$
and is defined as:
- $\ds \int_a^b \map f x \rd x = \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$
Integrand
In the expression for the definite integral:
- $\ds \int_a^b \map f x \rd x$
or primitive (that is, indefinite integral:
- $\ds \int \map f x \rd x$
the function $f$ is called the integrand.
Also known as
Many sources whose target consists of students at a relatively elementary level refer to this merely as a definite integral.
Expositions which delve deeper into the structure of integral calculus often establish the concepts of the Riemann integral and the Darboux integral, and contrast them with the Lebesgue integral, which is an extension of the concept into the more general field of measure theory.
Also see
There are more general definitions of integration; see: