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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{The Integral}

There are two uses of the term ``integral'' in calculus. One is as a method
of finding areas, extending the ``method of exhaustion'' described in the
Introduction. The other is as a synonym of the antiderivative of a function.
What \emph{should} be surprising is that these two ideas are intertwined, in
that you find areas by using antiderivatives. However, that connection is commonly
the only way in which areas are actually computed, so the power of the theorem
that gives the connection between them is often overlooked. But that theorem,
the Fundamental Theorem of Calculus, or FTC, is the foundation of the development
of calculus (hence the name, of course). The discovery of this result, independently
and approximately at the same time by both Issac Newton and \_\_ Leibnitz, is
the reason those two mathematicians are given credit as the inventors of calculus.
The basic ideas of the derivative were known before Newton and Leibnitz, and
the method of exhaustion goes back to classical Greek mathematicians, but the
connection between the two was the key to developing the analytical theory and
applications that we now call calculus.


\subsection{The definition of the integral}

The integral as ``area under a curve'' (the \emph{definite integral}) is defined,
and computed in the few cases where it can reasonably be found from the definition.
Properties of the definite integral are described


\subsection{The Fundamental Theorem}

Here we state and prove the FTC, which is the theoretical high point of the
course. 


\subsection{Indefinite integrals}

Since the FTC implies that the way to compute definite integrals involves antiderivatives,
which are usually called \emph{indefinite integrals}, here we find the first
methods to systematically compute them.


\subsection{Techniques of integration}

It is in general much more difficult to compute (indefinite) integrals than
derivatives. In this section we derive and apply the three most basic techniques
of integration, substitution, integration by parts, and partial fractions. there
are other techniques which we will not discuss, postponing them to the next
semester.

\end{document}
