Question

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.

$\mathit{x}\mathbf{-}{\mathbf{\text{cos}}}^{\mathbf{2}}\mathbf{\text{6}}\mathit{x}\mathbf{\text{on}}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{)}$

Short answer

The average value of the function$x-{\text{cos}}^2\text{6}x\text{on}\left(0,\frac{\pi }{6}\right)$ is calculated to be. It means that the area on the upper half of the curve is not equal to the lower half of the function

Step-by-step solution

Definition of amplitude, period, frequency, and velocity amplitude.

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

Given parameters

The given function is$x-{\cos }^{2}6x$.

The average value of function on interval$\left(0,\frac{\pi }{6}\right)$ is to be found.

Calculation of average value of function in given interval

Integrate given equation with upper limit be$\frac{\pi }{6}$and lower limit be 0. Use the formulae to calculate the average value of function f(x).

localid="1664265124905" $$\begin{gathered} f(x{)}_{Avg}=\frac{{\int }_0^{{\displaystyle \frac{\pi }{6}}}{\displaystyle \left(x-{\cos }^{2}6x\right)}{\displaystyle .}{\displaystyle d}{\displaystyle x}}{{\displaystyle \frac{\pi }{6}-0}} \\ f(x{)}_{Avg}=\frac{\pi }{6}\left|{\left(\frac{x^2}{2}\right)}_0^{\frac{\pi }{6}}-{\int }_0^{\frac{\pi }{6}}\frac{1+\cos 12x}{2}.dx\right| \\ f(x{)}_{Avg}=\frac{\pi }{6}\left\{\frac{{\pi }^2}{2\times 36}-0-{\left(\frac{x}{2}+\frac{\sin 12x}{24}\right)}_0^{\frac{\pi }{6}}\right\} \end{gathered}$$

Simplify further,

$$\begin{gathered} f(x{)}_{Avg}=\frac{\pi }{6}\left\{\frac{{\pi }^2}{72}-\frac{\pi }{12}+\left(\frac{1}{24}\sin \frac{\pi }{2}-0\right)\right\} \\ f(x{)}_{Avg}=\frac{\pi }{12}-\frac{1}{2}+\frac{1}{4\pi } \end{gathered}$$

Hence, the average value of the functionlocalid="1664265857317" $x-{\cos }^{2}6xon\left(0,\frac{\pi }{6}\right)$is $\frac{\pi }{12}-\frac{1}{2}+\frac{1}{4\pi }$.

Graph of the function

Use graphing calculator to graph of the function $x-{\cos }^{2}6xon\left(0,\frac{\pi }{6}\right)$.

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.