In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same ${\mathbf{\text{sin}}}^{\mathbf{2}}\mathbf{\text{3}}\mathit{x}\mathbf{\text{on}}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\text{4}}\mathit{\pi }\mathbf{)}$.

The average value of the function in the interval (a, b)is defined as.

$\mathit{f}\mathbf{(}\mathbf{x}{\mathbf{)}}_{\mathbf{A}\mathbf{v}\mathbf{g}}\mathbf{=}\frac{{\mathbf{\int }}_{\mathbf{a}}^{\mathbf{b}}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.}\mathbf{d}\mathbf{x}}{\mathbf{b}\mathbf{-}\mathbf{a}}$

Integration of sine function is$\mathbf{\int }\mathit{s}\mathit{i}\mathit{n}\left(ax+b\right)\mathbf{.}\mathit{d}\mathit{x}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{a}}\mathit{c}\mathit{o}\mathit{s}\left(ax+b\right)$.

The given function is${\sin }^{2}3x$.

The average value of function on interval$\left(0,4\pi \right)$ is to be found.

Integrate given equation with upper limit be $4\mathrm{\pi }$and lower limit be 0. Use the formulae to calculate the average value of function f(x) as follows:

$$\begin{gathered} f(x{)}_{Avg}=\frac{{\int }_0^{4\pi }\left(\frac{1-\cos 6x}{2}\right).dx}{4\pi -0} \\ f(x{)}_{Avg}=\frac{1}{8\pi }{\left(x-\left(\frac{-\sin 6x}{6}\right)\right)}_0^{4\pi } \\ f(x{)}_{Avg}=\frac{{\displaystyle 1}}{{\displaystyle 8\pi }}\left\{4\pi +\frac{1}{6}\sin 24\pi -0\right\} \\ f(x{)}_{Avg}=\frac{1}{2} \end{gathered}$$

Hence, the average value of the function${\sin }^{2}3xon\left(0,4\pi \right)$ is $\frac{1}{2}$.

Use graphing calculator to graph of the function ${\sin }^{2}3xon\left(0,4\pi \right)$.

Therefore, it is clear from the graph that the area subtends by the function above the X-axis is not equal to the area subtended below the X-axis.