$$ \newcommand{\R}{\mathbb{R}}
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\newcommand{\dx}{\text{ dx}}
\newcommand{\rang}{\text{rang}}
\newcommand{\s}{\ \ \ \ \ \ }
\newcommand{\arrows}{\s \Leftrightarrow \s}
\newcommand{\Arrows}{\s \Longleftrightarrow \s}
\newcommand{\arrow}{\s \Rightarrow \s}
\newcommand{\c}{\bcancel}
\newcommand{\v}[2]{
\begin{pmatrix}
#1 \\
#2 \\
\end{pmatrix}
}
\newcommand{\vt}[3]{
\begin{pmatrix}
#1 \\
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#3 \\
\end{pmatrix}
}
\newcommand{\stack}[2]{
\substack{
#1 \\
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}
}
\newcommand{\atom}[3]{
\substack{
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\ce{#3}
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$$
Arc Length
You can find the length of an arc by adding the small length of the infinity small vectors making up the arc.
$$\vec{r(t)} = f(t) i + g(t) j + h(t) k$$
$$|r'(t)| = \sqrt{f'(t)^{2} + g'(t)^{2} + h'(t)^{2}}$$
$$L=\int_{\alpha}^{\beta} |r'(t)| dt$$