# Definition:Definite Integral/Riemann

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## 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 $\closedint a b$ 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:

- $\ds \int_a^b \map f x \rd x$

More usually (and informally), we say:

**$f$ is (Riemann) integrable over $\closedint a b$**.

### Riemann Integral as Integral Operator

Let $C \closedint a b$ be the space of continuous functions.

Let $x \in C \closedint a b$ be a Riemann integrable function.

Let $\R$ be the set of real numbers.

The **Riemann integral operator**, denoted by $I$, is the mapping $I : C \closedint a b \to \R$ such that:

- $\ds \map I x := \int_a^b \map x t \rd t$

where $\ds \int_a^b \map x t \rd t$ is the Riemann integral.

## Also denoted as

The notation:

- $\ds \int_a^b f$

can occasionally be seen.

## Examples

### Definite Integral of $x^2$ from $0$ to $2$

- $\ds \int_0^2 x^2 \rd x = \dfrac 8 3$

### Definite Integral of $\sqrt x$ from $0$ to $4$

- $\ds \int_0^4 \sqrt x \rd x = \dfrac {16} 3$

### Definite Integral of $\cos x$ from $0$ to $\dfrac \pi 2$

- $\ds \int_0^{\pi / 2} \cos x \rd x = 1$

### Definite Integral of $\dfrac 1 x$ from $1$ to $e$

- $\ds \int_1^e \dfrac {\d x} x = 1$

### Definite Integral of $\dfrac 1 {1 - x}$ from $2$ to $3$

- $\ds \int_2^3 \dfrac {\d x} {1 - x} = \ln \dfrac 1 2$

## Also see

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

## Historical Note

Consider the Riemann sum:

- $\ds \sum_{i \mathop = 1}^n \map f {c_i} \Delta x_i$

Historically, the definite integral was an extension of this type of summation such that:

- The finite difference $\Delta x$ is instead the infinitely small difference $\rd x$

- The finite sum $\ds \Sigma$ is instead of the sum of an infinite number of infinitely small quantities: $\ds \int$

Hence the similarity in notation:

\(\ds \sum_a^b \map f x \Delta x\) | \(\to\) | \(\ds \int_a^b \map f x \rd x\) | as $\Delta x \to \rd x$ |

The notion of "infinitely small" does not exist in the modern formulation of real numbers.

Nevertheless, this idea is sometimes used as an informal interpretation of the Riemann integral.

Riemann established this definition in his paper *Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe* of $1854$, on the subject of Fourier series.

## Sources

- 1978: Garrett Birkhoff and Gian-Carlo Rota:
*Ordinary Differential Equations*(3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $2$ Fundamental Theorem of the Calculus - 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.): $\S 7.1$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Riemann integral, Riemann sum** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**integral**

- Riemann integral. V.A. Il'in (originator),
*Encyclopedia of Mathematics*. URL: https://www.encyclopediaofmath.org/index.php?title=Riemann_integral