\begin{frame} \frametitle{The Area below a Curve} Now let's look at a general \structure{curve above the $x$-axis}: \begin{block}{} The area below the curve of a function $f$ on an interval $[a,b]$. \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}); \only<-3>{ \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; } \only<4-13>{ \def\mwidth{3} \foreach \nrsteps/\mcolor in {3/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{\mstep} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {}; } } } \only<14->{ \def\mwidth{3} \foreach \nrsteps/\mcolor in {3/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{0} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {}; } } } \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (3.5,0) {$b$}; \end{scope} \end{tikzpicture} } \end{center}\vspace{-1ex} \pause We use $n$ rectangles: \mpause[6]{$\Delta x = \pause (b-a)/n$} \only<-7>{\begin{itemize} \item the width of the interval is $b-a$ \pause \item the width of each strip is $\Delta x = \pause (b-a)/n$ \pause \item the interval for the $i$-th strip is \pause $I_i = [a + (i-1)\Delta x,\;a+ i\Delta x]$ \end{itemize} } \pause\pause\pause\pause\pause\pause\medskip The area of the rectangles oriented at right-endpoints is:\vspace{-.5ex} \begin{talign} \alert{R_n} &= \mpause[1]{ \Delta x\cdot f(a + 1\Delta x) } \mpause{ + \Delta x\cdot f(a + 2\Delta x) } \mpause{ +\ldots+ \Delta x\cdot f(a + n\Delta x) } \\[-.5ex] &\mpause{= \Delta x\big(f(a + 1\Delta x) + f(a + 2\Delta x) + \ldots + f(a + n\Delta x)\big)} \end{talign} \pause\pause\pause\pause\pause The area of the rectangles oriented at left-endpoints is:\vspace{-.5ex} \begin{talign} \alert{L_n} &= \mpause[1]{ \Delta x\cdot f(a + 0\Delta x) } \mpause{ + \Delta x\cdot f(a + 1\Delta x) } \mpause{ +\ldots+ \Delta x\cdot f(a + (n-1)\Delta x) } \\[-.5ex] &\mpause{= \Delta x\big(f(a + 0\Delta x) + f(a + 2\Delta x) + \ldots + f(a + (n-1)\Delta x)\big)} \end{talign} \vspace{10cm} \end{frame}