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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Arc Length and Surface Area">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Tue Feb 11 23:48:05 1997}
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\section{Arc Length and Surface Area\label{arc length}}

\subsubsection{Arc Length}

\begin{theorem}
If a curve $C$ is described by the parametric equations $x=f\left( t\right) $%
, $y=g\left( t\right) $, $\alpha \leq t\leq \beta $, where $f\prime $ and $%
g\prime $ are continuous on $\left[ \alpha ,\beta \right] $ and $C$ is
traversed exactly once as $t$ increases from $\alpha $ to $\beta $, then the
length of $C$ is 
\[
L=\int_{\alpha }^{\beta }\sqrt{\left( \frac{dx}{dt}\right) ^{2}+\left( \frac{%
dy}{dt}\right) ^{2}}dt
\]
\end{theorem}

\subsubsection{Surface Area\label{surface area}}

If the curve given by the parametric equations $x=f\left( t\right) $, $%
y=g\left( t\right) $, $\alpha \leq t\leq \beta $, is rotated about the $x$%
-axis, where $f\prime $ and $g\prime $ are continuous and $g\left( t\right)
\geq 0$, then the area of the resulting surface is given by 
\[
S=\int_{\alpha }^{\beta }2\pi y\sqrt{\left( \frac{dx}{dt}\right) ^{2}+\left( 
\frac{dy}{dt}\right) ^{2}}dt 
\]
The general formulas $S=\int 2\pi yds$ and $S=\int 2\pi xds$ are still
valid, but for parametric curves you use 
\[
ds=\sqrt{\left( \frac{dx}{dt}\right) ^{2}+\left( \frac{dy}{dt}\right) ^{2}}%
dt 
\]

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