User:Keith.U/Definite Integral

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Definition: Definite Integral

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

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ 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 $\left[{a \,.\,.\, b}\right]$.


Suppose that:

$\displaystyle \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

where $\displaystyle \underline{\int_a^b}$ and $\displaystyle \overline{\int_a^b}$ denote the lower integral and upper integral, respectively.

Then $f$ is said to be (properly) Riemann integrable on $\left[{a \,.\,.\, b}\right]$.



More usually (and informally), we say:

$f$ is (Riemann) integrable over $\left[{a \,.\,.\, b}\right]$.


Definite Integral

Riemann Integral

The real number $L$ as defined above is called the Riemann integral of $f$ over $\left[{a \,.\,.\, b}\right]$ and is denoted:

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


Darboux Integral

The Darboux integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is denoted

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

and is defined as:

$\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x = \underline{\int_a^b} f \left({x}\right) \ \mathrm d x = \overline{\int_a^b} f \left({x}\right) \, \mathrm d x$


Integrand

In the expression for the definite integral:

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

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: