\begin{frame} \frametitle{The Area below a Curve} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \def\mwidth{3} \foreach \nrsteps/\mcolor in {3/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{\mstep} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \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 (3.5,0) {$b$}; \node at (1.8,-1) {right-endpoints with $5$ strips}; \end{scope} \end{tikzpicture}~\hspace{.5cm}~% \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \def\mwidth{3} \foreach \nrsteps/\mcolor in {3/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{0} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \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 (3.5,0) {$b$}; \node at (1.8,-1) {left-endpoints with $5$ strips}; \end{scope} \end{tikzpicture} }\\[1ex] \scalebox{.7}{ \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \def\mwidth{3} \foreach \nrsteps/\mcolor in {3/cred} { \def\mstep{\mwidth/(\nrsteps+1)} \def\mfunshift{\mstep/2} \foreach \xx in {0,...,\nrsteps} { \def\x{.5+ \xx*\mstep} \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 (3.5,0) {$b$}; \node at (1.8,-1) {midpoints with $5$ strips}; \end{scope} \end{tikzpicture} }\\[1ex] \scalebox{.7}{ \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \def\mwidth{3} \def\mstep{\mwidth/5} \def\mfunshift{0} \foreach \xx in {0,...,4} { \coordinate (height) at (0,0); \def\x{.5+ \xx*\mstep} \foreach \xxx in {0,0.01,...,1} { \gettikzxy{(height)} \def\mfunshift{\xxx*\mstep} \pgfmathparse{\mfun} \ifthenelse{\lengthtest{\pgfmathresult cm > \myy}}{ \coordinate (height) at (0,\pgfmathresult cm); }{} } \draw[thick,draw=cred!60!black,fill=cred,opacity=.5] ({\x},0) rectangle ($({\x+\mstep},0) + (height)$); } \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (3.5,0) {$b$}; \node at (1.8,-1) {upper sum with $5$ strips}; \end{scope} \end{tikzpicture}~\hspace{.5cm}~% \begin{tikzpicture}[default] \def\mfun{(4*(\x+\mfunshift) - 2.6*(\x+\mfunshift)^2 + .44*(\x+\mfunshift)^3)} {\def\diaborderx{.3cm} \def\diabordery{.3cm} \diagram[1]{-.5}{4}{-.4}{2}{1}} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=.5:3.5,samples=20] (\x,{\mfun}); \def\mwidth{3} \def\mstep{\mwidth/5} \def\mfunshift{0} \foreach \xx in {0,...,4} { \coordinate (height) at (0,5); \def\x{.5+ \xx*\mstep} \foreach \xxx in {0,0.01,...,1} { \gettikzxy{(height)} \def\mfunshift{\xxx*\mstep} \pgfmathparse{\mfun} \ifthenelse{\lengthtest{\pgfmathresult cm < \myy}}{ \coordinate (height) at (0,\pgfmathresult cm); }{} } \draw[thick,draw=cred!60!black,fill=cred,opacity=.5] ({\x},0) rectangle ($({\x+\mstep},0) + (height)$); } \node[anchor=north] at (.5,0) {$a$}; \node[anchor=north] at (3.5,0) {$b$}; \node at (1.8,-1) {lower sum with $5$ strips}; \end{scope} \end{tikzpicture} } \end{center}\vspace{-1ex} \end{frame}