# User:Keith.U/Definite Integral

## Contents

## 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: