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\begin{frame}
\frametitle{The Area below a Curve}

\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});
\def\mwidth{3}
\def\mstep{\mwidth/5}
\def\mfunshift{0}
\foreach \xx in {0,...,4} {
\coordinate (height) at (0,0);
\def\x{.5+ \xx*\mstep}
\foreach \xxx in {0,0.01,...,1} {
\gettikzxy{(height)}
\def\mfunshift{\xxx*\mstep}
\pgfmathparse{\mfun}
\ifthenelse{\lengthtest{\pgfmathresult cm > \myy}}{
\coordinate (height) at (0,\pgfmathresult cm);
}{}
}
\draw[thick,draw=cred!60!black,fill=cred,opacity=.5] ({\x},0) rectangle ($({\x+\mstep},0) + (height)$);
}
\node[anchor=north] at (.5,0) {$a$};
\node[anchor=north] at (3.5,0) {$b$};
\node at (2,-1.1) {upper sum $U_5$};
\end{scope}
\end{tikzpicture}~\hspace{.5cm}~%
\mpause[1]{%
\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});
\def\mwidth{3}
\def\mstep{\mwidth/5}
\def\mfunshift{0}
\foreach \xx in {0,...,4} {
\coordinate (height) at (0,5);
\def\x{.5+ \xx*\mstep}
\foreach \xxx in {0,0.01,...,1} {
\gettikzxy{(height)}
\def\mfunshift{\xxx*\mstep}
\pgfmathparse{\mfun}
\ifthenelse{\lengthtest{\pgfmathresult cm < \myy}}{
\coordinate (height) at (0,\pgfmathresult cm);
}{}
}
\draw[thick,draw=cred!60!black,fill=cred,opacity=.5] ({\x},0) rectangle ($({\x+\mstep},0) + (height)$);
}
\node[anchor=north] at (.5,0) {$a$};
\node[anchor=north] at (3.5,0) {$b$};
\node at (2,-1.1) {lower sum $D_5$};
\end{scope}
\end{tikzpicture}
}
}
\end{center}\vspace{-1ex}

\begin{block}{}
The \emph{upper sum} is
\begin{malign}
U_n = \Delta x\big(f(x_1) + f(x_2) + \ldots + f(x_n)\big)
\end{malign}\vspace{-1ex}

where $x_i$ is chosen from $I_i$ such that $f(x_i)$ is the maximum on $I_i$
\end{block}
\pause

\begin{block}{}
The \emph{lower sum} is
\begin{malign}
D_n = \Delta x\big(f(x_1) + f(x_2) + \ldots + f(x_n)\big)
\end{malign}\vspace{-1ex}

where $x_i$ is chosen from $I_i$ such that $f(x_i)$ is the minimum on $I_i$
\end{block}
\end{frame}