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\begin{frame}
  \frametitle{Motivation and Overview}

  \scalebox{.8}{
  \begin{tikzpicture}[default]
    \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)}

    \def\diax{\text{time}}
    \def\diay{\text{speed}}
    \def\diaborderx{.75cm}
    \def\diabordery{.75cm}
    \diagram[1]{-1}{4}{-1}{8}{1}
    \diagramannotatez
    \diagramannotatexx{1/1h,2/2h,3/3h}
    \diagramannotateyy{1/10 \text{mph},2/20 \text{mph},3/30 \text{mph},3/30 \text{mph}}
    \def\mfunshift{0}
    \draw[cred] plot[smooth,domain=0:4,samples=20] (\x,{\mfun});

    \node[rectangle,rounded corners=2mm,fill=yellow!15,draw=black,thin,align=left,drop shadow,inner sep=1.5mm] at (1.5,6)
      {\parbox{.52\textwidth}{
       What is the area under a curve?
       \begin{itemize}
         \item approximate?
         \item compute precisely?
       \end{itemize}
       }
      };
  

    \begin{scope}[xshift=.65\textwidth]
      \def\diax{\text{time}}
      \def\diay{\text{distance}}
      \def\diaborderx{.75cm}
      \def\diabordery{.75cm}
      \diagram[1]{-1}{4}{-1}{8}{1}
      \diagramannotatez
      \diagramannotatexx{1/1h,2/2h,3/3h}
      \diagramannotateyy{1/10 \text{mi},2/20 \text{mi},3/30 \text{mi},4/40 \text{mi},5/50 \text{mi},6/60 \text{mi},7/70 \text{mi}}

      \coordinate (z) at (0,0);
    \end{scope}


    \def\mwidth{4}
    \setcounter{slide}{2}
    \foreach \nrsteps/\mcolor in {3/cred,7/cblue,15/cgreen} {
      \setcounter{roundcounter}{\arabic{slide}}
      \def\mstep{\mwidth/(\nrsteps+1)}
      \def\mfunshift{\mstep}
      \coordinate (p) at (z);
      \foreach \xx in {0,...,\nrsteps} {
        \def\x{\xx*\mstep}
        \setcounter{tmpcount}{\arabic{roundcounter}}
        \addtocounter{tmpcount}{\nrsteps}
        \onslide<\arabic{slide}-\arabic{tmpcount}>{ 
          \draw[draw=none,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); 
          \node[include=\mcolor] at ({\x+\mfunshift},{\mfun}) {};
        }
        \onslide<\arabic{slide}->{ 
          \begin{scope}[shift=(z)]
            \coordinate (p') at ($(p) + ({\mstep},{\mstep*\mfun})$);
            \node[\mcolor,include] at (p') {};
            \draw[\mcolor] (p) -- (p');
            \coordinate (p) at (p');
          \end{scope}
        }
        \addtocounter{slide}{1}
      }
    }
    
    \onslide<30->{
      \begin{scope}[shift=(z)]
        \draw[line width=3mm,cblue!5,opacity=.5] plot[smooth,domain=0:4,samples=20] (\x,{2*\x^2 - 2.6/3*\x^3 + .44/4*\x^4)});
        \draw[line width=1mm,orange] plot[smooth,domain=0:4,samples=20] (\x,{2*\x^2 - 2.6/3*\x^3 + .44/4*\x^4)});
      \end{scope}
  
      \node[rectangle,rounded corners=2mm,fill=yellow!15,draw=black,thin,align=left,drop shadow,inner sep=1.5mm] at (5,-1)
        {The finer approximations get closer and closer to the precise solution \tikz[baseline=-.5ex] \draw[line width=1mm,orange] (0,0) -- (1,0);.};
    }
  \end{tikzpicture}
  }

\end{frame}