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| Let $I=[a,b]$ be a closed interval, $f: I \rightarrow \mathbb{R}$ be bounded on $I$, $n \in \mathbb{N}$, and $P = \{[x_0, x_1), [x_1, x_2), \dots [x_{n-1}, x_n]\}$ be a partition of $I$. The \emph{Riemann sum} of $f$ over $I$ with respect to the partition $P$ is defined as |
Suppose there is a function $f: I \rightarrow R$ where $I=[a,b]$ is a closed interval, and $f$ is bounded on $I$. If we have a finite set of points $\{x_0, x_1, x_2, \dots x_n\}$ such that $a = x_0 < x_1 < x_2 \dots < x_n = b$, then this set creates a \PMlinkescapeword{partition} partition $P = \{[x_0, x_1), [x_1, x_2), \dots [x_{n-1}, x_n]\}$ of $I$. |
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If $P$ is a partition with $n \in \mathbb{N}$ elements of $I$, then the Riemann sum of $f$ over $I$ with the partition $P$ is defined as |
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$$S=\sum_{j=1}^n f(c_j)(x_j-x_{j-1})$$
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$$S = \sum_{i=1}^{n} f(y_i)(x_{i}-x_{i-1})$$
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| where $c_j \in [x_{j-1},x_j]$ is chosen arbitrary. |
where $x_{i-1} \leq y_i \leq x_i$. The choice of $y_i$ is arbitrary. If $y_i = x_{i-1}$ for all $i$, then $S$ is called a left Riemann sum. If $y_i = x_i$, then $S$ is called a right Riemann sum. |
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Suppose we have |
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| If $c_j=x_{j-1}$ for all $j$, then $S$ is called a \emph{left Riemann sum}. |
$$S = \sum_{i=1}^{n} b(x_{i}-x_{i-1})$$ |
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| If $c_j=x_j$ for all $j$, then $S$ is called a \emph{\PMlinkescapetext{right} Riemann sum}. |
where $b$ is the supremum of $f$ over $[x_{i-1}, x_{i}]$; then $S$ is defined to be an upper Riemann sum. Similarly, if $b$ is the infimum of $f$ over $[x_{i-1}, x_{i}]$, then $S$ is a lower Riemann sum. |
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| Equivalently, the Riemann sum can be defined as |
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| $$S=\sum_{j=1}^n b_j(x_j-x_{j-1})$$ |
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| where $b_j \in \{ f(x):x\in[x_{j-1},x_j]\}$ is chosen arbitrarily. |
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| If $\displaystyle b_j=\sup_{x\in[x_{j-1},x_j]} f(x)$, then $S$ is called an \emph{upper Riemann sum}. |
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| If $\displaystyle b_j=\inf_{x\in[x_{j-1},x_j]} f(x)$, then $S$ is called a \emph{lower Riemann sum}. |
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