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Let $I\subseteq\mathbb{R}^n$ be a box, $P$ a partition of $I$, and $f:I\to\mathbb{R}$ a function. Let the lower Riemann sum of $f$ with respect to $P$ be $L(f, P)$, and the upper Riemann sum of $f$ with respect to $P$
be $U(f, P)$
. Then the lower Riemann sum of $f$ with respect to $P$ is
$$\begin{align*}
\underline{\int_I}f(\vec{x})d\vec{x}:=\sup_{P\in\mathbb{P}_I}\{L(f, P)\}
\end{align*}$$ and the upper Riemann integral of $f$ with respect to $P$ is
$$\begin{align*}
\bar{\int_I}f(\vec{x})d\vec{x}:=\inf_{P\in\mathbb{P}_I}\{U(f, P)\}
\end{align*}$$
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Let $I\subseteq\mathbb{R}^n$ be a box, $P$ a partition of $I$, and $f:I\to\mathbb{R}$ a function. Let the lower Riemann sum of $f$ with respect to $P$ be $L(f, P)$, and the upper Riemann sum of $f$ with respect to $P$
be $U(f, P)$
. Then the lower Riemann sum of $f$ with respect to $P$ is
$$\begin{align*}
\underline{\int_I}f(\vec{x})d\vec{x}:=\sup_{P\in\mathbb{P}_I}\{L(f, P)\}
\end{align*}$$ and the upper Riemann integral of $f$ with respect to $P$ is
$$\begin{align*}
\bar{\int_I}f(\vec{x})d\vec{x}:=\inf_{P\in\mathbb{P}_I}\{U(f, P)\}
\end{align*}$$
Concepts
Coming soonUsed In
Coming soonHypothesis
Coming soonResults
Coming soon
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