$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Properties of the Riemann Integral

Let $S\subseteq\mathbb{R}^n$ be nonempty and bounded and let $f, g:S\to\mathbb{R}$ be integrable on $S$.
  1. For any $\alpha, \beta\in\mathbb{R}$, the function $h:=\alpha f+\beta g$ is integrable on $S$ and $$\begin{align*} \int_S\alpha f(\vec{x})+\beta g(\vec{x})d\vec{x}=\alpha\int_S f(\vec{x}) d\vec{x} + \beta\int_S g(\vec{x}) d\vec{x} \end{align*}$$
  2. If $f(\vec{x})\leq g(\vec{x})$ for all $\vec{x}\in S$, then $$\begin{align*} \int_S f(\vec{x})d\vec{x} \leq \int_S g(\vec{x}) d\vec{x} \end{align*}$$
    3. The function $|f|$ is integrable on $S$ and $$\begin{align*} \int_S |f(\vec{x})|d\vec{x} \geq \left|\int_S f(\vec{x})d\vec{x}\right| \end{align*}$$
  3. If $S$ has content, then $$\begin{align*} m\mu(S)\leq \int_S f(\vec{x})d\vec{x} \leq M\mu(S) \end{align*}$$ where $m$ and $M$ are respectively lower and upper bounds of $f$ on $S$.

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Properties of the Riemann Integral

Let $S\subseteq\mathbb{R}^n$ be nonempty and bounded and let $f, g:S\to\mathbb{R}$ be integrable on $S$.
  1. For any $\alpha, \beta\in\mathbb{R}$, the function $h:=\alpha f+\beta g$ is integrable on $S$ and $$\begin{align*} \int_S\alpha f(\vec{x})+\beta g(\vec{x})d\vec{x}=\alpha\int_S f(\vec{x}) d\vec{x} + \beta\int_S g(\vec{x}) d\vec{x} \end{align*}$$
  2. If $f(\vec{x})\leq g(\vec{x})$ for all $\vec{x}\in S$, then $$\begin{align*} \int_S f(\vec{x})d\vec{x} \leq \int_S g(\vec{x}) d\vec{x} \end{align*}$$
    3. The function $|f|$ is integrable on $S$ and $$\begin{align*} \int_S |f(\vec{x})|d\vec{x} \geq \left|\int_S f(\vec{x})d\vec{x}\right| \end{align*}$$
  3. If $S$ has content, then $$\begin{align*} m\mu(S)\leq \int_S f(\vec{x})d\vec{x} \leq M\mu(S) \end{align*}$$ where $m$ and $M$ are respectively lower and upper bounds of $f$ on $S$.

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FullPage
result
concepts
hypothesis
implications
proof

Properties of the Riemann Integral

Let $S\subseteq\mathbb{R}^n$ be nonempty and bounded and let $f, g:S\to\mathbb{R}$ be integrable on $S$.
  1. For any $\alpha, \beta\in\mathbb{R}$, the function $h:=\alpha f+\beta g$ is integrable on $S$ and $$\begin{align*} \int_S\alpha f(\vec{x})+\beta g(\vec{x})d\vec{x}=\alpha\int_S f(\vec{x}) d\vec{x} + \beta\int_S g(\vec{x}) d\vec{x} \end{align*}$$
  2. If $f(\vec{x})\leq g(\vec{x})$ for all $\vec{x}\in S$, then $$\begin{align*} \int_S f(\vec{x})d\vec{x} \leq \int_S g(\vec{x}) d\vec{x} \end{align*}$$
    3. The function $|f|$ is integrable on $S$ and $$\begin{align*} \int_S |f(\vec{x})|d\vec{x} \geq \left|\int_S f(\vec{x})d\vec{x}\right| \end{align*}$$
  3. If $S$ has content, then $$\begin{align*} m\mu(S)\leq \int_S f(\vec{x})d\vec{x} \leq M\mu(S) \end{align*}$$ where $m$ and $M$ are respectively lower and upper bounds of $f$ on $S$.

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Properties of the Riemann Integral

Let $S\subseteq\mathbb{R}^n$ be nonempty and bounded and let $f, g:S\to\mathbb{R}$ be integrable on $S$.
  1. For any $\alpha, \beta\in\mathbb{R}$, the function $h:=\alpha f+\beta g$ is integrable on $S$ and $$\begin{align*} \int_S\alpha f(\vec{x})+\beta g(\vec{x})d\vec{x}=\alpha\int_S f(\vec{x}) d\vec{x} + \beta\int_S g(\vec{x}) d\vec{x} \end{align*}$$
  2. If $f(\vec{x})\leq g(\vec{x})$ for all $\vec{x}\in S$, then $$\begin{align*} \int_S f(\vec{x})d\vec{x} \leq \int_S g(\vec{x}) d\vec{x} \end{align*}$$
    3. The function $|f|$ is integrable on $S$ and $$\begin{align*} \int_S |f(\vec{x})|d\vec{x} \geq \left|\int_S f(\vec{x})d\vec{x}\right| \end{align*}$$
  3. If $S$ has content, then $$\begin{align*} m\mu(S)\leq \int_S f(\vec{x})d\vec{x} \leq M\mu(S) \end{align*}$$ where $m$ and $M$ are respectively lower and upper bounds of $f$ on $S$.

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