Example: Animation for Upper Riemann Sum

Published 2020-11-01 | Author: Edson José Teixeira

An animation to the area of calculation using the upper Riemann sum.

That’s an approximation of an integral by a finite sum, named after the German mathematician Riemann. It is calculated by partitioning the region below the the curve into rectangles and summarizing their areas. To get a better approximation, the region is devided more finely. As the rectangles get smaller, the Rieman sum approaches the Riemann integral. This animation shows it.

Download the PDF and play it, to see it actually animated, for example using the Adobe Reader.

Download as: [PDF] [TEX]

Animation for Upper Riemann Sum

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% Animation for Upper Riemann Sum
% Author: Edson José Teixeira
\documentclass[10pt]{beamer}
\usepackage[controls]{animate}
\usepackage{tikz}
\usetikzlibrary{arrows}
% Beamer Settings
\usetheme{Warsaw}
% Counters
\newcounter{higher} 
\setcounter{higher}{1}
\begin{document}
\begin{frame}[fragile]{Upper Riemann Sum}
  \begin{figure}
    \begin{animateinline}[poster = first, controls]{5}
      \whiledo{\thehigher<30}{
        \begin{tikzpicture}[line cap=round, line join=round, >=triangle 45,
                            x=4.0cm, y=1.0cm, scale=1]
          \draw [->,color=black] (-0.1,0) -- (2.5,0);
          \foreach \x in {1,2}
            \draw [shift={(\x,0)}, color=black] (0pt,2pt)
                  -- (0pt,-2pt) node [below] {\footnotesize $\x$};
          \draw [color=black] (2.5,0) node [below] {$x$};
          \draw [->,color=black] (0,-0.1) -- (0,4.5);
          \foreach \y in {1,2,3,4}
            \draw [shift={(0,\y)}, color=black] (2pt,0pt)
                  -- (-2pt,0pt) node[left] {\footnotesize $\y$};
          \draw [color=black] (0,4.5) node [right] {$y$};
          \draw [color=black] (0pt,-10pt) node [left] {\footnotesize $0$};
          \draw [domain=0:2.2, line width=1.0pt] plot (\x,{(\x)^2});
          \clip(0,-0.5) rectangle (3,5);
          \draw (2,0) -- (2,4);
          \foreach \i in {1,...,\thehigher}
            \draw [fill=black,fill opacity=0.3, smooth,samples=50] ({1+(\i-1)/\thehigher},{(1+(\i)/\thehigher)^2})
                  --({1+(\i)/\thehigher},{(1+(\i)/\thehigher)^2})
                  --  ({1+(\i)/\thehigher},0)
                  -- ({1+(\i-1)/\thehigher},0)
                  -- cycle;
        \end{tikzpicture}
        %
        \stepcounter{higher}
        \ifthenelse{\thehigher<30}{ \newframe }{\end{animateinline} }
      }
      \caption{Upper Riemann Sum}
  \end{figure}
\end{frame}
\end{document}

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