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# Riemann integral

Let $I=[a,b]$ be an interval of $\mathbb{R}$ and let $f\colon I\to\mathbb{R}$ be a bounded function. For any finite set of points $\{x_{0},x_{1},x_{2},\ldots,x_{n}\}$ such that $a=x_{0}<x_{1}<x_{2}\cdots<x_{n}=b$, there is a corresponding partition $P=\{[x_{0},x_{1}),[x_{1},x_{2}),\ldots,[x_{{n-1}},x_{n}]\}$ of $I$.

Let $C(\epsilon)$ be the set of all partitions of $I$ 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 $f$ is Riemann-integrable over $I$, and the Riemann integral of $f$ over $I$ is defined by

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

## Mathematics Subject Classification

28-00*no label found*26A42

*no label found*

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## Attached Articles

example of a non Riemann integrable function by paolini

a lecture on the partial fraction decomposition method by alozano

left hand rule by Wkbj79

right hand rule by Wkbj79

midpoint rule by Wkbj79

example of estimating a Riemann integral by Wkbj79

integral over plane region by pahio

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## Additional Reference

PlanetMath article: Non-Newtonian calculus.