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 to denote a Riemann integral, but this is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Definite Integral of $2 x$ from $2$ to $3$

$\ds \int_2^3 2 x \rd x = 5$


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

  • Results about Riemann integrals can be found here.


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

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