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Revision difference : Riemann integral |
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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 $\{x_0, x_1, x_2, \dotsc, x_n\}$ 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 $I$.
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Suppose there is a function $f\colon D \rightarrow R$ where $D, R \subseteq \mathbb{R}$ and that there is a closed interval $I = [a,b]$ such that $I \subseteq D$. For any finite set of points $\{x_0, x_1, x_2, \dots x_n\}$ such that $a = x_0 < x_1 < x_2 \dots < x_n = b$, there is a corresponding partition $P = \{[x_0, x_1), [x_1, x_2), \dotsc, [x_n-1, x_n]\}$ of $I$.
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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 \PMlinkname{decreasing}{IncreasingdecreasingmonotoneFunction} and $S_{*}(\epsilon)$ is \PMlinkname{increasing}{IncreasingdecreasingmonotoneFunction}. Therefore, the limits $S^{*}=\lim_{\epsilon\to 0} S^{*}(\epsilon)$ and $S_{*}=\lim_{\epsilon\to 0} S_{*}(\epsilon)$ exist. If $S^{*} = S_{*}$, then $f$ is Riemann-integrable over $I$, and the Riemann integral of $f$ over $I$ is defined by
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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^{*} $S^{*}=\lim_{\epsilon\to 0} S^{*}(\epsilon)$ and $S_{*}=\lim_{\epsilon\to 0} S_{*}(\epsilon)$ exist. If $S^{*} = S_{*}$, then $f$ is Riemann-integrable over $I$, and the Riemann integral of $f$ over $I$ is defined by
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| \begin{equation*} |
\begin{equation*} |
| \int_{a}^{b} f(x)dx = S^{*} = S_{*}. |
\int_{a}^{b} f(x)dx = S^{*} = S_{*}. |
| \end{equation*} |
\end{equation*} |
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