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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}

\begin{example}
Find the areas enclosed by the curves \( y=x^{2} \) and \( y=2x+1 \). 
\end{example}

\subsubsection*{Solution}

The region is like this:

\vspace{0.3cm}
{\par\centering \includegraphics{area-3.ps} \par}
\vspace{0.3cm}

Byt {}``the area enclosed by{}'' or {}``the region bounded by{}'' we would
mean the region trapped between those two curves, that name-brand {}``swoosh{}''
region. To find the intersection points, which in this case define the first
and last \( x \) you need for this region, you have to find where the curves
have the same \( y \) for the same \( x \), where \[
x^{2}=2x+1,\]
which by the quadratic formula is at \[
x=1\pm \sqrt{2},\]
where \( x^{2}-2x-1=0 \). 

Then, note that the top curve is the curve \( y=2x+1 \), and the bottom is
\( y=x^{2} \), in the region bounded by these two curves. By the formula, then,\begin{eqnarray*}
Area & = & \int _{1-\sqrt{2}}^{1+\sqrt{2}}\left( (2x+1)-x^{2}\right) dx\\
 & = & \left. x^{2}+x-\frac{x^{3}}{3}\right| _{1-\sqrt{2}}^{1+\sqrt{2}}\\
 & = & \left( 1+\sqrt{2}\right) ^{2}+\left( 1+\sqrt{2}\right) -\frac{\left( 1+\sqrt{2}\right) ^{3}}{3}\\
 &  & -\left( \left( 1-\sqrt{2}\right) ^{2}+\left( 1-\sqrt{2}\right) -\frac{\left( 1-\sqrt{2}\right) ^{3}}{3}\right) \\
 & = & \frac{8}{3}\sqrt{2}\\
 & \cong  & 3.77
\end{eqnarray*}

\end{document}
