%% LyX 1.1 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,oneside]{amsart}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength\parskip{\medskipamount}
\setlength\parindent{0pt}

\makeatletter


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
\providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 \theoremstyle{plain}    
 \newtheorem{thm}{Theorem}[section]
 \numberwithin{equation}{section} %% Comment out for sequentially-numbered
 \numberwithin{figure}{section} %% Comment out for sequentially-numbered
 \theoremstyle{definition}
  \newtheorem{example}[thm]{Example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{5}
\setcounter{subsection}{1}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The Fundamental Theorem of Calculus}

\begin{example}
\[
\int _{2}^{5}f(x)dx.\]
 where \[
f(x)=\left\{ \begin{array}{cc}
3x-4, & \mathrm{if}\, x\geq 4\\
x+2, & \mathrm{if}\, x<4
\end{array}\right. \]
 
\end{example}

\subsubsection*{Solution}

Split the integral into regions where we can unambiguously describe the function:\textit{\begin{eqnarray*}
\int _{2}^{5}f(x)dx & = & \int _{2}^{4}f(x)dx+\int _{4}^{5}f(x)dx\\
 & = & \int _{2}^{4}x+2dx+\int _{4}^{5}3x-4dx\\
 & = & x^{2}/2+2x|_{2}^{4}+3x^{2}/2-4x|_{4}^{5}\\
 & = & (4^{2}/2+2(4))-(2^{2}/2+2(2))+(3(5^{2})/2-4(5))-(3(4^{2})/2-4(4))\\
 & = & 39/2.
\end{eqnarray*}
}Verify for yourself that the antidifferentiation is correct..
\end{document}
