\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{block}{}
    Let $f$ be continuous on $[a,b]$, and let 
    \begin{itemize}
      \item $x_1 < x_2 < \ldots < x_n$ be all $x$-intercepts in $[a,b]$,
      \item define $x_0 = a$ \;and $x_{n+1} = b$
    \end{itemize}
    Then the area between the curve $f$ and the $x$-axis from $a$ to $b$ is
    \begin{talign}
      A = \int_a^b |f(x)|dx \;=\; \sum_{i=0}^{n}\left|\int_{x_i}^{x_{i+1}} \!\!\!f(x)\,dx\right|
    \end{talign}\vspace{-1.5ex}
  \end{block}
    
  
  \vspace{10cm}
\end{frame}