PlanetMath: Riemann integral

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Riemann integral

(Definition)

Let be an interval of $\mathbb{R}$ and let $f\colon I\to \mathbb{R}$ be a bounded function. For any finite set of points $\{x_0, x_1, x_2, \dotsc, x_n\}$ such that $a = x_0 < x_1 < x_2 \dotsb < x_n = b$ , there is a corresponding partition $P = \{[x_0, x_1), [x_1, x_2), \dotsc, [x_{n-1}, x_n]\}$ of .

Let $C(\epsilon)$ be the set of all partitions of with $\max (x_{i+1}-x_i)<\epsilon$ . Then let $S^{*}(\epsilon)$ be the infimum of the set of upper Riemann sums with each partition in $C(\epsilon)$ , and let $S_{*}(\epsilon)$ be the supremum of the set of lower Riemann sums with each partition in $C(\epsilon)$ . If $\epsilon_1<\epsilon_2$ , then $C(\epsilon_1)\subset C(\epsilon_2)$ , so $S^{*}(\epsilon)$ is decreasing and $S_{*}(\epsilon)$ is increasing. Moreover, $\lvert S^{*}(\epsilon)\rvert$ and $\lvert S_{*}(\epsilon)\rvert$ are bounded by $(b-a)\sup_x \lvert f(x)\rvert$ . Therefore, the limits $S^{*}=\lim_{\epsilon\to 0} S^{*}(\epsilon)$ and $S_{*}=\lim_{\epsilon\to 0} S_{*}(\epsilon)$ exist and are finite. If $S^{*} = S_{*}$ , then is Riemann-integrable over , and the Riemann integral of over is defined by

$\displaystyle \int_{a}^{b} f(x)dx = S^{*} = S_{*}.$

"Riemann integral" is owned by bbukh. [ full author list (2) | owner history (1) ]

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See Also: Riemann sum, Lebesgue integral

Also defines:

Riemann integrable

Attachments:: continuous functions are Riemann integrable (Theorem) by paolini
example of a non Riemann integrable function (Example) by paolini
a lecture on the partial fraction decomposition method (Feature) by alozano
left hand rule (Theorem) by Wkbj79
right hand rule (Theorem) by Wkbj79
midpoint rule (Theorem) by Wkbj79
example of estimating a Riemann integral (Example) by Wkbj79

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Cross-references: finite, limits, bounded, lower Riemann sums, supremum, upper Riemann sums, infimum, partition, points, finite set, bounded function, interval
There are 36 references to this entry.

This is version 9 of Riemann integral, born on 2001-10-19, modified 2006-06-10.
Object id is 370, canonical name is RiemannIntegral.
Accessed 15583 times total.

Classification:

AMS MSC:	28-00 (Measure and integration :: General reference works )
	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

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