\frametitle{The Definite Integral}
%     Let $f$ be a function defined on $[a,b]$.
%     \medskip
    The \emph{definite integral of $f$ from $a$ to $b$} is
      \alert{\int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x}
    provided that the limit exists, and has the same value for all possible choices
    of the \emph{sample points} 
      $x_i$ from the interval $[a + (i-1)\Delta x,\; a + i\Delta x]$ 
    where $\Delta x = \frac{b-a}{n}$.\pause\medskip
    If the limit exists, we call $f$ \emph{integrable} on $[a,b]$.
  The procedure of calculating an integral is called \emph{integration}.
  Here $a$ is the \emph{lower limit} and $b$ is the \emph{upper limit} of integration.
    The sum \quad \alert{$\sum_{i = 1}^n f(x_i) \Delta x$} \quad 
    is called \emph{Riemann sum}.