\begin{frame} \frametitle{Area Between Curves} The definite integral can be interpreted as the \emph{net area}, that is: \begin{minipage}{.6\textwidth} \begin{center} \scalebox{.9}{ \begin{tikzpicture}[default] \def\mfun{(-.9 + (\x-3+\mfunshift)^2 - .1*(\x-3+\mfunshift)^4)} \diagram[1]{-.5}{6}{-1}{1.7}{1} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=.5:2,samples=100] (\x,{\mfun}) -- (.5,0) -- cycle; \draw[fill=cred,draw=none,opacity=.5] plot[smooth,domain=2:4,samples=100] (\x,{\mfun}) -- cycle; \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=4:5.5,samples=100] (\x,{\mfun}) -- (5.5,0) -- cycle; \draw[cred] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}); \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (5.5,0) {$b$}; \node[scale=1.8] at (.9,.5) {+}; \node[scale=1.8] at (5.15,.5) {+}; \node[scale=3] at (3,-.5) {-}; \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{minipage}{.39\textwidth} \begin{talign} \int_a^b f(x)dx\\ \end{talign} \end{minipage} \pause \begin{alertblock}{} What if we want the area between the curve and the $x$-axis? \end{alertblock}\smallskip \begin{minipage}{.6\textwidth} \begin{center} \scalebox{.9}{ \begin{tikzpicture}[default] \def\mfun{(-.9 + (\x-3+\mfunshift)^2 - .1*(\x-3+\mfunshift)^4)} \diagram[1]{-.5}{6}{-1}{1.7}{1} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=.5:2,samples=100] (\x,{\mfun}) -- (.5,0) -- cycle; \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=2:4,samples=100] (\x,{\mfun}) -- cycle; \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=4:5.5,samples=100] (\x,{\mfun}) -- (5.5,0) -- cycle; \draw[cred] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}); \mpause[2]{ \draw[cred,dashed] plot[smooth,domain=2:4,samples=100] (\x,{-\mfun}); } \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (5.5,0) {$b$}; \node[scale=1.8] at (.9,.5) {+}; \node[scale=1.8] at (5.15,.5) {+}; \node[scale=1.8] at (3,-.5) {+}; \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{minipage}{.39\textwidth} \pause \begin{talign} \int_a^b |f(x)|dx\\ \end{talign} \end{minipage} \vspace{10cm} \end{frame}