Resources for Applications of Integration of Trig Functions
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Questions
13
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Video Tutorials
2
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Applications of Integration of Trig Functions Theory
![Applications of the integration of trigonometric functions involves, given the derivative function then determine the original function and finding the areas involving trigonometric graphs.\\ \begin{multicols}{2} \textbf{Example 1}\\%10590 \(\dfrac{{dy}}{{dx}} = 3\sin x + {\sec ^2}x\), find \(y\) in terms of \(x\) if \(y=4\) when \(x=0\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \frac{d y}{d x}=3 \sin x+\sec ^{2} x \rightarrow y &=-3 \cos x+\tan x+c \\ \text { when } y=4, x=0 \quad 4 &=-3 \cos 0+\tan 0+c \\ 4 &=-3+c \rightarrow c=7 \\ \therefore\ y &=-3 \cos x+\tan x+7 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %10587 Find the area under the curve \(y = 2\cos x\) and the \(x\)-axis as indicated in the following diagram. \\ \begin{center} \begin{tikzpicture}[ declare function={a(\x)=2*cos(deg(x));} ] \def \domain{0:2*pi+0.2} \def \xmax{2*pi+pi/4} \def \xmin{-0.6} \def \ymax{3} \def \ymin{-3} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={Stealth-Stealth,very thick}, grid=both, %major, ylabel = $\ylabel$, xlabel = $\xlabel$, width=3.5in, height=2.5in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, minor x tick num=1, minor y tick num=1, axis line style = thick, major tick style = thick, minor tick style = thick, xtick distance = 3, xlabel style={right}, ytick distance = 2, ylabel style={above}, x grid style={thin, opacity=0.8}, y grid style={thin, opacity=0.8}, axis on top=false, xtick={pi/4,pi/2,3*pi/4,pi,5*pi/4,6*pi/4,7*pi/4,2*pi}, ytick={-3,-2,...,3}, xticklabels={ $\frac{\pi}{4}$, $\frac{\pi}{2}$, $\frac{3\pi}{4}$, $\pi$, $\frac{5\pi}{4}$, $\frac{3\pi}{2}$, $\frac{7\pi}{4}$, $2\pi$}, % Labels for x ticks % extra x ticks={0}, % yticklabels={$-1$, $-\frac{1}{2}$, $0$, $\frac{1}{2}$, $1$}, % Labels for y ticks extra x tick style={xticklabel style={anchor=north east}} % enlargelimits=true, clip=false, ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax); % Plot the sine function \addplot[name path=a, ultra thick, ->, samples=300, smooth, domain=\domain, red] {a(x)} node [pos=0.9, left, red, font=\small] {}; \addplot[name path=b] {0}; \addplot[color=green!40,opacity=0.6] fill between[of=a and b,soft clip={domain=pi/2:3*pi/2}]; \end{axis} \end{tikzpicture} %\includegraphics[width=0.3\textwidth]{6779251c-d0ee-457f-9a3c-c10ba6d795f5} \end{center} \textbf{Example 2 solution}\\ $\begin{aligned} \text{To find the}\ &\text{x-intercepts}\\ \text{let}\ 2 \cos x&=0\\ \therefore x &=\frac{\pi}{2} \text { or } \frac{3 \pi}{2} \\ \text { Area } &=\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} |2 \cos x| d x\\ &=\left[|2 \sin x \mid\right]_{\frac{\pi}{2}}^{\frac{3 \pi}{2}}\\ \text { Area } &=2\left[\left|\sin \frac{3 \pi}{2}-\sin \frac{\pi}{2}\right|\right] \\ &=2[|-1-1|] \\ &=4 u^{2} \end{aligned}$ \end{multicols}](/media/5bnpvamy/4793.png)