\begin{frame}
  \frametitle{The Definite Integral}
  
  \begin{block}{}
%     Let $f$ be a function defined on $[a,b]$.
%     \medskip
%     
    The \emph{definite integral of $f$ from $a$ to $b$} is
    \begin{talign}
      \alert{\int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x}
    \end{talign}
    provided that the limit exists, and has the same value for all possible choices
    of the \emph{sample points} 
    \begin{center}
      $x_i$ from the interval $[a + (i-1)\Delta x,\; a + i\Delta x]$ 
    \end{center}
    where $\Delta x = \frac{b-a}{n}$.\pause\medskip
    
    If the limit exists, we call $f$ \emph{integrable} on $[a,b]$.
  \end{block}
  \pause\smallskip
  
  The procedure of calculating an integral is called \emph{integration}.
  \pause\medskip
  
  Here $a$ is the \emph{lower limit} and $b$ is the \emph{upper limit} of integration.
  \pause
  
  \begin{block}{}
    The sum \quad \alert{$\sum_{i = 1}^n f(x_i) \Delta x$} \quad 
    is called \emph{Riemann sum}. 
  \end{block}
  \vspace{10cm}
\end{frame}