\begin{frame} \frametitle{The Area below a Curve} \begin{block}{} The \emph{area} $A$ under the graph of a continuous function $f$ whose graph lies \structure{above the $x$-axis} is the limit: % is the limit of the sum of the areas of the approximating rectangles: \begin{talign} A &= \lim_{n\to \infty} R_n \\ &= \lim_{n\to \infty} \left[ \Delta x\big(f(a + 1\Delta x) + f(a + 2\Delta x) + \ldots + f(a + n\Delta x)\big) \right] \end{talign} where $\Delta x = (b-a)/n$. \end{block} \begin{center} \scalebox{.9}{ \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \draw[draw=none,fill=cred,opacity=.5] (.5,0) -- plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}) -- (3.5,0) -- (.5,0) -- cycle; \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (3.5,0) {$b$}; \end{scope} \end{tikzpicture} } \end{center}\vspace{-1ex} \pause \begin{block}{} For continuous $f$ this limit always exists, and is the same as \begin{talign} \lim_{n\to \infty} L_n = \lim_{n\to \infty} \left[ \Delta x\big(f(a + 0\Delta x) + \ldots + f(a + (n-1)\Delta x)\big) \right] \end{talign} \end{block} \vspace{10cm} \end{frame}