AreaAproximation

\documentclass[compress,9pt]{beamer}

\usepackage{tikz}   %TikZ is required for this to work.  Make sure this exists before the next line

\begin{document}
	
	\section{Area aproximation: Rectangles + Triangles }
	\begin{frame}[fragile]{Area $\approxeq$ Rectangles + Triangles}
		\begin{figure}[h]
			\centering
			\pgfmathsetmacro{\circleRadius}{4}
			\resizebox{0.5\textwidth}{!}{
				\begin{tikzpicture}
				% Draw X axis
				\draw[thick,->] (0,0) -- (\circleRadius+0.3,0) node[anchor=north west]{$t$};
				% Draw X axis labels
				\foreach \x in {0,...,\circleRadius}
				{
					\draw (\x,0) -- (\x,-3pt) node[anchor=north] {$t_\x$};
				}
				% Draw Y axis
				\draw[thick,->] (0,0) -- (0,\circleRadius+0.3) node[anchor=south east]{$\omega$};
				% Draw a quarter of circle
				\draw[red,thick] (0,0\circleRadius) arc[start angle=90, end angle=0, radius=\circleRadius];
				% Draw trapeziums
				\foreach \thisX in {1,...,\circleRadius}
				{
					% Calculate rectangle points
					\pgfmathsetmacro{\thisAngle}{acos((\thisX)/\circleRadius)}
					\pgfmathsetmacro{\thisY}{\circleRadius*sin(\thisAngle)}
					% Calculate triangle points
					\pgfmathsetmacro{\previousAngle}{acos((\thisX-1)/\circleRadius)}
					\pgfmathsetmacro{\previousY}{\circleRadius*sin(\previousAngle)}
					% Draw triangle
					\draw[dashed, fill=blue!30] (\thisX-1,\previousY) -- (\thisX-1,\thisY) -- (\thisX,\thisY) -- cycle;
					% Draw rectangle
					\draw[fill=gray!30] (\thisX-1,0) rectangle (\thisX,\thisY);
					% draw y axis labels
					\draw (0,\thisY) -- (-3pt,\thisY) node[anchor=east] {$\omega_\thisX$};
				}
				\end{tikzpicture}
			}
		\end{figure}	
		
		\begin{equation}\label{eq:omega2phiTrapezIni}
		\boldsymbol{\theta}_k \approx \boldsymbol{\theta}_{k-1} + (t_k - t_{k-1}) \boldsymbol{\omega}_{k-1} + \frac{1}{2} (t_k - t_{k-1}) (\boldsymbol{\omega}_{k} - \boldsymbol{\omega}_{k-1}),\quad k \ge 1
		\end{equation}
	\end{frame}
	
		
\end{document}