\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} \pause\bigskip Note every function is integrable. \pause\bigskip However, most of the functions we work with are: \pause \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} \vspace{10cm} \end{frame}