Question

What is the average value of the function `f(x)=cos(8x)` on the interval `[0,pi/2]`?

Answer 1

`0`

Explanation:

The average value of a function `f` over the interval `(a,b)` is given by `frac{int_a^b f(x) dx}{b-a}`. For example, average velocity is given by `frac{int_0^tau v(t) dt}{tau}`.

So in this case

`frac{int_0^{pi/2} f(x) dx}{pi/2-0} = frac{int_0^{pi/2}cos(8x)dx}{pi/2}`

`= frac{[sin(8x)/8]_0^{pi/2}}{pi/2}`

`= frac{[sin(4pi)/8-sin(0)/8]}{pi/2}`

`= frac{[0/8-0/8]}{pi/2}`

`= 0`