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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\section{Applications of Integration}

The general idea of applications of integration is to reduce the computation,
the measurement, you need to do into several simpler pieces. On each little
piece, the measurement can be reasonably approximated by some simplified method,
and the errors in that approximation are small, getting smaller as the size
of the pieces get smaller. The total measurement is the sum of the pieces, and
as the size of each piece gets smaller (and the total number of pieces gets
larger) the approximations go to the exact measurement in the limit. 

That sketch actually describes integration itself, but it can be applied to
any situation where the whole is the sum of the parts.


\subsection{Area between curves}

This first application is a lot like the integral itself, except that the emphasis
is now on actually finding the physical area. The regions are more complicated
as well, being any region in the plane bounded by curves.


\subsection{Average Value}

The average of a discrete set of values is easy to define, but the average of
a function, though intuititvely clear, requires a limit to make complete sense.
Here some manipulation is needed to make the summation process actually work
out to be an integral.


\subsection{Volumes}

Volumes of regions in space are one of the standard applications of integration.
There are several standard methods of breaking down the problem into simplified
pieces, and they lead to different formulas. It is the process of decomposition
and turning into an integral that is the crux of the method, however.


\subsection{Arclength}

The length of a curve is another application that requires some manipulation
to make the limiting process actually become an integral. Many of the problems
are specially formulated, since finding the arclength of many simple curves
leads to integrals that can't be solved in terms of elementary functions.


\subsection{Work}

This is a more physical application, but illustrates the method of decomposition
that underlies all applications of integration.

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