\begin{frame} \frametitle{Area Between Curves} \begin{exampleblock}{} Find the area of the region bounded by the curves \vspace{-.5ex} \begin{talign} y = \sin x && y = \cos x && x = 0 && x = \pi/2 \end{talign} \pause Area is equal to the area between $\sin x - \cos x$ and the $x$-axis:\vspace{-.5ex} \begin{talign} A = \int_0^{\pi/2} |\sin x - \cos x|dx \end{talign} \pause We have to find the $x$-intercepts in the interval $[0,\pi/2]$:\vspace{-.5ex} \begin{talign} \sin x - \cos = 0 \mpause[1]{\;\iff\; \sin x = \cos x} \mpause{\;\iff\; x =} \mpause{ \pi/4} \end{talign} \pause\pause\pause\pause Antiderivative of $f(x) = \sin x - \cos x$ is $F(x) = \pause -\cos x - \sin x$:\vspace{-.5ex} \pause \only<-15>{ \begin{talign} \alert{A} &= \mpause[1]{ \left| \int_0^{\pi/4} f(x)dx \right| + \left| \int_{\pi/4}^{\pi/2} f(x)dx \right| } \mpause{ = | F(x) \big]_0^{\pi/4} | + | F(x) \big]_{\pi/4}^{\pi/2} | } \\[-.5ex] \mpause{ &= | (-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}) - (-1 - 0)| + | (-0 - 1) - (-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}})| }\\ \mpause{ &= | -\frac{2}{\sqrt{2}} +1| + | -1 + \frac{2}{\sqrt{2}}| } \mpause{ = \sqrt{2} -1 + -1 + \sqrt{2} } \mpause{ \alert{= 2\sqrt{2} -2} } \end{talign} } \only<16>{ \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,xscale=2] \def\mfuna{(cos (180/pi *\x))} \def\mfunb{(sin (180/pi *\x))} \diagram[1]{-.5}{3}{-1}{1.2}{1} \diagramannotatez \def\mfunshift{0} \begin{scope}[ultra thick] \draw[cred] plot[smooth,domain=-.5:3,samples=100] (\x,{\mfuna}) node[above] {$f(x)$}; \draw[cblue] plot[smooth,domain=-.5:3,samples=100] (\x,{\mfunb}) node[above,yshift=1.5mm] {$g(x)$}; \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=0:pi/4,samples=100] (\x,{\mfuna}) -- plot[smooth,domain=pi/4:0,samples=100] (\x,{\mfunb}) -- cycle; \draw[fill=cgreen,draw=none,opacity=.5] plot[smooth,domain=pi/4:pi/2,samples=100] (\x,{\mfuna}) -- plot[smooth,domain=pi/2:pi/4,samples=100] (\x,{\mfunb}) -- cycle; \end{scope} \end{tikzpicture} } \end{center}\vspace{-2.5ex} \begin{talign} \alert{A = 2\sqrt{2} -2} \end{talign} } \end{exampleblock} \end{frame}