\begin{frame} \frametitle{Area Between Curves} \begin{alertblock}{} What if we want the area between the curve and the $x$-axis? \end{alertblock}\smallskip \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}); \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (5.5,0) {$b$}; \node[scale=1.8] at (.9,.5) {+}; \node at (.9,.9) {$A_1$}; \node[scale=1.8] at (5.15,.5) {+}; \node at (5.15,.9) {$A_3$}; \node[scale=1.8] at (3,-.6) {+}; \node at (3,-.25) {$A_2$}; \draw[gray] (2,.2) -- node[at end,below,black] {$x_1$} (2,-.2); \draw[gray] (4,.2) -- node[at end,below,black] {$x_2$} (4,-.2); \end{scope} \end{tikzpicture} } \end{center}\vspace{-.5ex} \begin{exampleblock}{} For example, let us consider the diagram above. \medskip \pause % Let $x_1 < x_2$ be the $x$-intercepts. % \medskip % \pause % The area between the curve and the $x$-axis from $a$ to $b$ is \begin{talign} A = \int_a^b |f(x)|dx \;=\; \left| \int_a^{x_1} \!\!\!f(x)\,dx \right| \;+\; \left| \int_{x_1}^{x_2} \!\!\!f(x)\,dx \right| \;+\; \left| \int_{x_2}^b \!\!\!f(x)\,dx \right| \end{talign}\vspace{-1ex} \pause \alert{Note that we split the integral from $a$ to the first $x$-intercept, from the first to the second $x$-intercept,\ldots} \end{exampleblock} \vspace{10cm} \end{frame}