\begin{frame} \frametitle{The Definite Integral} \begin{exampleblock}{} Evaluate the Riemann sum for \begin{talign} f(x) = 2x-5 \end{talign} from $0$ to $6$ using $3$ strips and right endpoints as sample points. \pause\medskip We have: \begin{itemize} \pause \item the width of the strips is $\Delta x = \pause (6-0) / 3 \pause = 2$ \pause \item the intervals of the strips are \pause $[0,2]$, $[2,4]$, $[4,6]$ \pause \item the right endpoints are \pause $x_1 = 2$, $x_2=4$, $x_3 = 6$ \pause \item the values at $x_i$'s are \pause $f(x_1) = -1$\pause, $f(x_2) = 3$\pause, $f(x_3) = 7$ \end{itemize} \pause\medskip Thus the Riemann sum using $3$ strips and right endpoints is: \begin{talign} R_3 = \sum_{i=1}^3 f(x_i)\cdot \Delta x = \mpause[1]{2\cdot (-1)} \mpause{+ 2\cdot 3} \mpause{+ 2\cdot 7} \mpause{= 18} \end{talign} \end{exampleblock} \end{frame}