%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Iterated Integrals">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:10:09 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=Reference.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{<LINK REL="contents" HREF="..\Reference\ReferenceIndex.tex#Reference Index">}
%TCIDATA{<LINK REL="next" HREF="cal12_03.tex">}
%TCIDATA{<LINK REL="parent" HREF="CAL_contents.tex">}
%TCIDATA{<LINK REL="previous" HREF="cal12_01.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{Iterated Integrals\label{Iterated Integrals}}

\subsubsection{Fubini's Theorem\label{Fubini's Theorem}}

\begin{theorem}
If $f$ is continuous on the rectangle 
\[
R=\left\{ (x,y)\left| a\leq x\leq b,c\leq y\leq d\right. \right\} 
\]
then 
\[
\iint_{R}f(x,y)dA=\int_{a}^{b}\int_{c}^{d}f(x,y)dydx=\int_{c}^{d}%
\int_{a}^{b}f(x,y)dxdy
\]
More generally, this is true if we assume that $f$ is bounded on $R$, $f$ is
discontinuous only on a finite number of smooth curves, and the iterated
integrals exist.
\end{theorem}

\begin{remark}
In the special case where $f(x,y)$ can be factored as the product of a
function of $x$ only and a function of $y$ only, the double integral of $f$
can be written in a particularly simple form. To be specific, suppose that $%
f(x,y)=g(x)h(y)$ and $R=\left[ a,b\right] \times c,d\rbrack $. Then Fubini's
Theorem gives

\[
\iint_{R}f(x,y)dA=\int_{c}^{d}\int_{a}^{b}g(x)h(x)dxdy=\int_{c}^{d}\left[
\int_{a}^{b}g(x)h(y)dx\right] dy=\int_{a}^{b}g(x)dx\int_{c}^{d}h(y)dy
\]
\end{remark}

\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}}

\noindent \hrulefill

\begin{center}
\begin{tabular}{ccc}
{\small \hyperref{\FRAME{itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Computing Techniques}{}{}{%
../../Help/DMindex.tex}} & {\small \hyperref{\FRAME{itbpF}{13.9375pt}{%
14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
13.9375pt;height 14.4375pt;depth 3pt;original-width
24.0625pt;original-height 24.8125pt;cropleft "0";croptop "0.9959";cropright
"0.9937";cropbottom "0";filename 'Nblogo.wmf';file-properties "XNPEU";}}%
General Information}{}{}{../../Help/Index.tex}} & {\small \hyperref{\FRAME{%
itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Reference Library}{}{}{%
../ReferenceIndex.tex}}
\end{tabular}
\end{center}

\noindent \hrulefill

\end{document}
