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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Double Integrals in Polar Coordinates">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:10:11 1997}
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\section{Double Integrals In Polar Coordinates\label%
{Double Integrals in Polar Coordinates}}

\subsubsection{Change to Polar Coordinates In a Double Integral\label%
{Change to Polar coordinates in a Double Integral}}

If $f$ is continuous on a polar rectangle $R$ given by $0\leq a\leq r\leq b$
and $\alpha \leq \theta \leq \beta $, where $0\leq \beta -\alpha \leq 2\pi $%
, then 
\begin{eqnarray*}
\iint_{R}f(x,y)dA &=&\int_{\alpha }^{\beta }\int_{a}^{b}f(r\cos \theta
,r\sin \theta )rdrd\theta  \\
&&
\end{eqnarray*}
If $f$ is continuous on a polar region of the form 
\[
D=\left\{ (r,\theta )\left| {}\right. \alpha \leq \theta \leq \beta
,h_{1}\left( \theta \right) \leq r\leq h_{2}\left( \theta \right) \right\} 
\]
then 
\[
\iint_{D}f(x,y)dA=\int_{\alpha }^{\beta }\int_{h_{1}(\theta )}^{h_{2}(\theta
)}f(r\cos \theta ,r\sin \theta )rdrd\theta 
\]

\subsection{Related Reference Topics}

{\small \hyperref{Polar coordinates, geometry}{}{}{%
GE06_01.tex#Polar coordinates}}

{\small \hyperref{Polar coordinates, calculus}{}{}{%
Cal08_04.tex#polar coordinate system}}

\subsection{Related Scientific Notebook Topics}

\vspace{1pt}{\small \hyperref{Iterated integrals}{}{}{%
../../Help/DM7-8.tex#Iterated integrals}}

{\small \hyperref{Order of integration}{}{}{../../Help/DM7-8.tex#Order of
integration}}

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