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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{Area between curves}

The \textit{area} between curves \( y=f(x) \) and \( y=g(x) \), for, say \( x=a \)
up to \( x=b \), as pictured, 

\vspace{0.3cm}
{\par\centering \includegraphics{area.ps} \par}
\vspace{0.3cm}

is found in the following process: Break the domain \( [a,b] \) into a bunch
of subintervals, each of width \( \Delta x \). Over each little piece, say
centered at the point \( x_{i} \), the sliver of area between the curves is
a thin \textit{almost} rectangular region, of area \[
\Delta A_{i}=(g(x_{i})-f(x_{i}))\Delta x,\]
 approximately. 

\vspace{0.3cm}
{\par\centering \includegraphics{area-1.ps} \par}
\vspace{0.3cm}

The total area, then, is \[
\sum _{i=1}^{n}(g(x_{i})-f(x_{i}))\Delta x=\sum _{i=1}^{n}\Delta A_{i},\]


at least approximately. Of course, as the width \( \Delta x \) of the slices
gets smaller, the approximation to the true area improves, so that the exact
area \( A \) is: \[
A=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}(f(x_{i})-g(x_{i}))\Delta x,\]


which works out to be the same limit as the integral: \[
\int _{a}^{b}(f(x)-g(x))dx,\]
so this integral measures the area between the curves. Note that, as with the
picture, the endpoints \( x=a \) and \( x=b \) may or may not be places where
the curves intersect.  

Note that here the integral is an exact value for the area, which is approximated
by the sums leading up to the integral. That's the general idea with applications
of integration; you approximate a situation with geometrically reasonable models,
and as the approximation gets finer and finer, two things happen:  

\begin{enumerate}
\item The approximations approach the real object to be measured, and 
\item The formula becomes the integral, the limit of Riemann sums, of some function. 
\end{enumerate}
\begin{example}
Find the areas enclosed by the curves \( y=x^{2} \) and \( y=2x+1 \). 
\end{example}

\subsubsection*{Solution}

\begin{xca}
Find the areas enclosed by the curves \( y=x \) and \( y=x^{3} \) (this one
has a bit of a catch). 
\end{xca}
\vspace{0.3cm}
{\par\centering \includegraphics{area-4.ps} \par}
\vspace{0.3cm}

\begin{example}
Find the areas enclosed by the curves \( y=\arctan (x) \) and \( y=\pi x/4 \).
\textit{The constants are supposed to help.} 
\end{example}

\subsubsection*{Solution}

\begin{xca}
Find the areas enclosed by the curves \[
y=\sqrt{4-x^{2}},\, y=x,\, \textrm{and}\, y=-x.\]
\end{xca}

\end{document}
