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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{thm*}
\textbf{\textup{{[}Second Fundamental Theorem of Calculus{]}.}} Let \( f \)
be a continuous function on an open interval containing \( [a,b] \). Let \( F(x) \)
be \textbf{any} antiderivative of \( f \). Then:\[
\int _{a}^{b}f(x)dx=F(b)-F(a):=\left. F(x)\right| ^{b}_{a}.\]

\end{thm*}
\textbf{\emph{Proof:}}

Set \( G(x) \) as before to be \[
G(x):=\int _{a}^{x}f(t)dt.\]
 Then, by the First FTC, \( G'(x)=f(x) \). On the other hand, we \textit{assume}
that \( F'(x)=f(x) \). A corollary to the MVF says that two functions with
the same derivative can only differ by a constant, so \( F(x)+C=G(x) \). Thus,\begin{eqnarray*}
\int _{a}^{b}f(x)dx & = & G(b)\\
 & = & G(b)-G(a)\, \, (\mathrm{since}\, G(a)=0)\\
 & = & (F(b)+C)-(F(a)+C)\\
 & = & F(b)-F(a)=\left. F(x)\right| ^{b}_{a}^{b}_{a.}.
\end{eqnarray*}

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