\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}