\begin{frame} \frametitle{The Area below a Curve} \begin{exampleblock}{} Estimate the area below the curve $f(x) = x^2$ from $0$ to $1$. \end{exampleblock} \medskip \begin{minipage}{.49\textwidth} \begin{center} \scalebox{.65}{ \begin{tikzpicture}[default] \def\mfun{(4*((\x+\mfunshift)/4)^2)} \diagram[1]{-.5}{4.5}{-.4}{4.5}{1} \diagramannotatez \diagramannotatexx{1/.25,2/.5,3/.75,4/1} \diagramannotateyy{1/.25,2/.5,3/.75,4/1} \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=0:4,samples=20] (\x,{\mfun}); \def\mwidth{4} \setcounter{slide}{1} \foreach \nrsteps/\mcolor in {3/cred,7/cred,19/cred} { \setcounter{roundcounter}{\arabic{slide}} \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{0} \foreach \xx in {0,...,\nrsteps} { \def\x{\xx*\mstep} \setcounter{tmpcount}{\arabic{roundcounter}} \addtocounter{tmpcount}{\nrsteps} \onslide<\arabic{slide}>{ \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {}; } } \addtocounter{slide}{1} } \end{scope} \end{tikzpicture} } \end{center} \end{minipage}~ \begin{minipage}{.49\textwidth} \begin{center} \scalebox{.65}{ \begin{tikzpicture}[default] \def\mfun{(4*((\x+\mfunshift)/4)^2)} \diagram[1]{-.5}{4.5}{-.4}{4.5}{1} \diagramannotatez \diagramannotatexx{1/.25,2/.5,3/.75,4/1} \diagramannotateyy{1/.25,2/.5,3/.75,4/1} \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=0:4,samples=20] (\x,{\mfun}); \def\mwidth{4} \setcounter{slide}{1} \foreach \nrsteps/\mcolor in {3/cred,7/cred,19/cred} { \setcounter{roundcounter}{\arabic{slide}} \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{\mstep} \foreach \xx in {0,...,\nrsteps} { \def\x{\xx*\mstep} \setcounter{tmpcount}{\arabic{roundcounter}} \addtocounter{tmpcount}{\nrsteps} \onslide<\arabic{slide}>{ \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {}; } } \addtocounter{slide}{1} } \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{talign} 0.21875 = L_4 \hspace{.5cm}<\hspace{.5cm} &A \hspace{.5cm}<\hspace{.5cm} R_4 = 0.46875 \\ \mpause[1]{ 0.2734375 = L_8 \hspace{.5cm}<\hspace{.5cm} &A \hspace{.5cm}<\hspace{.5cm} R_8 = 0.3984375 }\\ \mpause{ 0.3087500 = L_{20} \hspace{.5cm}<\hspace{.5cm} &A \hspace{.5cm}<\hspace{.5cm} R_{20} = 0.3587500 } % \mpause{ % 0.3234000 = L_{50} \hspace{.5cm}<\hspace{.5cm} &A \hspace{.5cm}<\hspace{.5cm} R_{50} = 0.3434000 % } % \mpause{ % 0.3283500 = L_{100} \hspace{.5cm}<\hspace{.5cm} &A \hspace{.5cm}<\hspace{.5cm} R_{100} = 0.3383500 % } \end{talign}\vspace{-3ex} \begin{exampleblock}{} We have obtained an estimation of $A$: \begin{itemize} \item we can improve the estimation by taking more strips \end{itemize} \end{exampleblock} \vspace{10cm} \end{frame}