\frametitle{The Definite Integral}

  If the limit
    \int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x
  exists, then $f$ is called \emph{integrable} on $[a,b]$.
  Note every function is integrable.
  However, most of the functions we work with are:
      \item $f$ is continuous on $[a,b]$, or
      \item $f$ has only a finite number of jump discontinuities,
    then $f$ is integrable on $[a,b]$,
    that is, the $\int_{a}^{b} f(x)dx$ exist.