\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}