\begin{frame}
  \frametitle{The Definite Integral}
 
  \begin{block}{}
  If $f$ is integrable on $[a,b]$, then the limit
  \begin{talign}
    \int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x
  \end{talign}
  gives the same value no matter how we choose the sample points $x_i$
  from the $i$-th interval.
  \end{block}
  \pause\bigskip
 
  Thus for simplicity we can choose the right end points.
  \pause\bigskip
  
  This simplifies the definition of the definite integral:
  \begin{block}{}
    If $f$ is integrable on $[a,b]$, then
    \begin{talign}
      \int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x
    \end{talign}
    where \quad$\Delta x = \frac{b-a}{n}$\quad and \quad $x_i = a + i\Delta$.
  \end{block}
  \vspace{10cm}
\end{frame}