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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{The Fundamental Theorem of Calculus}

\begin{example}
\[
\int _{0}^{2}\sqrt{4-x^{2}}dx\]
 
\end{example}

\subsubsection*{Solution: }

Here the integral itself is much harder (which is what I wanted you to see).
But, the question becomes finding the area of a quarter-circle of radius 2,
since that is the graph \[
y=\sqrt{4-x^{2},}\]
 for \( x\in [0,2] \). The answer is \( \pi  \). 

This is not {}``cheating{}'', since in the final analysis the integral is
that area under the curve. Using integration techniques is the \emph{usual}
way we evaluate that area, but it is not the only possible way. In this case
it is much easier to solve it this way, and to use integration techniques requires
a considerable build-up (which is part of the next course).
\end{document}
