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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Areas and Lengths in Polar Coordinates">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Feb 12 00:01:33 1997}
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\section{Areas and Lengths In Polar Coordinates}

\subsubsection{Area of a Polar Region\label{area of a polar region}}

To find the formula for the area of a region whose boundary is described by
a polar equation, let $R$ be the region bounded by the polar curve $%
r=f\left( \theta \right) $ and by the rays $\theta =a$ and $\theta =b$,
where $f$ is a positive continuous function and $0<b-a\leq 2\pi $. Given the
formula for the area of a sector of a circle 
\[
A=\frac{1}{2}r^{2}\theta 
\]
the formula for the area $A$ of the polar region $R$ is 
\[
A=\int_{a}^{b}\frac{1}{2}\left[ f\left( \theta \right) \right] ^{2}d\theta 
\]
This formula is often written 
\[
A=\int_{a}^{b}\frac{1}{2}r^{2}d\theta 
\]
where $r=f\left( \theta \right) $.

\subsubsection{Arc Length\label{arc length}}

The length of a curve with polar equation $r=f\left( \theta \right) $, $%
a\leq \theta \leq b$, is 
\[
L=\int_{a}^{b}\sqrt{r^{2}+\left( \frac{dr}{d\theta }\right) ^{2}}d\theta 
\]

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