\begin{frame} \frametitle{The Definite Integral} \begin{block}{} The sum \quad \alert{$\sum_{i = 1}^n f(x_i) \Delta x$} \quad is called \emph{Riemann sum}. \end{block} \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[cred] plot[smooth,domain=.5:5.5,samples=100] (\x,{\mfun}); % \only<-3>{ % \draw[draw=none,fill=cred,opacity=.5] (.5,0) -- plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}) -- (3.5,0) -- (.5,0) -- cycle; % } \def\mwidth{5} \foreach \nrsteps/\mcolor in {6/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{.5*\mstep} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \pgfmathparse{{\mfun}} \ifthenelse{\lengthtest{\pgfmathresult cm > 0cm}}{ \def\mcolor{cgreen} }{} \draw[thick,draw=\mcolor!60!black,fill=\mcolor,opacity=.5] ({\x},0) rectangle ({\x+\mstep},{\mfun}); \node[include=\mcolor] at ({\x+\mfunshift},{\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}\vspace{-1.5ex} \begin{block}{} The \emph{Riemann sum} is the sum of the area of rectangles above the $x$-axis (the green ones) \alert{minus} the sum of the area of the rectangles below the $x$-axis (the red ones). \end{block} \pause \begin{block}{} The sample points $x_i$ can be arbitrary from the $i$-th interval: \begin{itemize} \item left endpoints, right endpoints or middle of the interval, or \item at maximum (upper sum), or at minimum (lower sum). \end{itemize} \end{block} \end{frame}