Question

To find the average value of the function on the given interval.

$\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\frac{\mathbf{7}\mathbf{\pi x}}{\mathbf{2}}\mathbf{}\mathit{o}\mathit{n}\mathbf{}\mathbf{(}\mathbf{0}\mathbf{,}\frac{\mathbf{8}}{\mathbf{7}}\mathbf{)}$.

Short answer

The average value of the given function over two periods is $\frac{1}{2}$.

Step-by-step solution

Given

The given function on the interval is ${\cos }^2\frac{7\pi x}{2}on\left(0,\frac{8}{7}\right)$.

The concept of the average value of a function over a particular interva

The average value of a function over a particular interval can be found with an expression involving an integral.

Let's say that interval is$\mathbf{[}\mathit{a}\mathbf{,}\mathit{b}\mathbf{|}$.

Then the average value off(x)over said interval is$\frac{\mathbf{1}}{\mathbf{b}\mathbf{-}\mathbf{a}}{\mathbf{\int }}_{\mathbf{a}}^{\mathbf{b}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}$.

Averagefunction on the given interval

From the given information, the average value is zero since all the functions are periodic with periods$T=\frac{8}{7}$.

The average function on the given interval is shown below.

$={\int }_0^{8/7}{\cos }^2\frac{7\pi x}{2}dx$

The period of the function

The period of the function that is integrated is:

$$\begin{gathered} \frac{2\pi }{T}=\frac{7\pi }{2} \\ T=4/7 \end{gathered}$$

Thus, the average value of the given function over two periods is $\frac{1}{2}$.