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\begin{frame}
\frametitle{The Definite Integral}

\begin{block}{}
If the limit
\begin{talign}
\int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x
\end{talign}
exists, then $f$ is called \emph{integrable} on $[a,b]$.
\end{block}
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Note every function is integrable.

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However, most of the functions we work with are:
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\begin{block}{}
If
\begin{itemize}
\item $f$ is continuous on $[a,b]$, or
\item $f$ has only a finite number of jump discontinuities,
\end{itemize}
then $f$ is integrable on $[a,b]$,
that is, the $\int_{a}^{b} f(x)dx$ exist.
\end{block}
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\end{frame}