Question

Question #8fab4

Answer 1

Please see below.

Explanation:

We'll need:

The average value of integrable function `f` from `x=a` to `x=b` (or "on `[a,b]`") is

`1/(b-a) int_a^b f(x) dx`,

and

`int 1/(1+x^2) dx = arctan x +C`,

and some trigonometry for special angles.

The average value of `f(x) = 2/(1+x^2)` on `[0,sqrt3]` is

`1/(sqrt3 - 0) int_0^sqrt3 2/(1+x^2) dx = {:2/sqrt3 arctanx| :}_0^sqrt3`

`= 2/sqrt3(arctansqrt3 - arctan0)`

`= 2/sqrt3(pi/3) = (2pi)/(3sqrt3)`