\begin{frame}
  \frametitle{The Area below a Curve}

  \begin{block}{}
  The area under a curve $f$ above the $x$-axis from $a$ to $b$ is:
  \begin{talign}
    A = \lim_{n\to \infty} \left( \sum_{i = 1}^n \Delta x\cdot f(x_i) \right)
  \end{talign}  
  where:
  \begin{itemize}
    \item $\Delta x = (b-a)/n$ is the width of the strips,
    \item $I_i = [a + (i-1)\Delta x,\;a+ i\Delta x]$ is the interval of the $i$-th strip,
    \item $x_i$ is the sample point from the $i$-th interval $I_i$.
  \end{itemize}
  \end{block} 
  \pause
  \medskip
  
  \begin{block}{}
    Usual choices for $x_i$ are
    \begin{itemize}
    \pause
      \item left endpoint $x_i = a + (i-1)\Delta x$ of the interval 
    \pause
      \item right endpoint $x_i = a + i\Delta x$ of the interval 
    \pause
      \item middle $x_i = a + (i-\frac{1}{2})\Delta x$ of the interval 
    \pause
      \item upper sum: $f(x_i)$ is the maximum on the interval $I_i$ 
    \pause
      \item lower sum: $f(x_i)$ is the minimum on the interval $I_i$ 
    \end{itemize}
  \end{block}
\end{frame}