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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Area of a Surface of Revolution">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Mon Mar 31 10:41:32 1997}
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\section{Area of a Surface of Revolution\label%
{area of a surface of revolution}}

\begin{itemize}
\item  The surface area of the surface obtained by rotating the curve $%
y=f\left( x\right) $, $a\leq x\leq b$, about the $x$-axis is 
\[
S=\int_{a}^{b}2\pi y\sqrt{1+\left( \frac{dy}{dx}\right) ^{2}}dx
\]

\item  If the curve is described as $x=g\left( y\right) $, $c\leq y\leq d$,
then the formula for the surface area is 
\[
S=\int_{c}^{d}2\pi y\sqrt{1+\left( \frac{dx}{dy}\right) ^{2}}dy
\]

\item  Both of these formulas can be summarized using the notation for 
\hyperref{arc length}{}{}{cal06_07.tex#arc length formula} as

\begin{enumerate}
\item  $S=\int 2\pi yds$ for a rotation about the $x$-axis,

\item  $S=\int 2\pi xds$ for a rotation about the $y$-axis,
\end{enumerate}
\end{itemize}

where you can use $ds=\sqrt{1+\left( \frac{dy}{dx}\right) ^{2}}dx$ or $ds=%
\sqrt{1+\left( \frac{dx}{dy}\right) ^{2}}dy$.

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Definite integrals}{}{}{../../Help/DM7-6.tex#Definite
integral}} 

\noindent \hrulefill

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