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

  Recall that the interval of the $i$-th strip is: 
  \begin{talign}
    I_i = [a + (i-1)\Delta x,\;a+i\Delta x]
  \end{talign}\vspace{-3ex}
  \pause
  
  \begin{exampleblock}{}\vspace{-.75ex}
  \begin{malign}
    R_n = \Delta x\big(f(x_1) + f(x_2) + \ldots + f(x_n)\big)
  \end{malign}\vspace{-1ex}
  
  where $x_i$ is the right endpoint of the interval $I_i$
  \end{exampleblock}
  \pause
  
  \begin{exampleblock}{}\vspace{-.75ex}
  \begin{malign}
    L_n = \Delta x\big(f(x_1) + f(x_2) + \ldots + f(x_n)\big)
  \end{malign}\vspace{-1ex}
  
  where $x_i$ is the left endpoint of the interval $I_i$
  \end{exampleblock}
  \pause\medskip

  \begin{block}{}
  For continuous curve $f$ above the $x$-axis we have:
  \begin{talign}
    A = \lim_{n\to \infty} \left[ \Delta x\big(f(x_1) + f(x_2) + \ldots + f(x_n)\big) \right]
  \end{talign}
  independent of what \emph{sample points} $x_i$ we take from $I_i$.
  \end{block}
  \pause
  The limit is the same no matter what $x_i$ we choose from $I_i$!
  \pause\medskip
  
  A famous choice of sample points are upper and lower sums\ldots
\end{frame}