Question

Find the average value of the function on the given interval?

Answer 1

`2/(5pi)`

Explanation:

The formula for the average value of a function `f(x)` on the interval `[a,b]` is:
`1/(b-a)int_a^bf(x)\ dx`

Applying this formula, we get:
`1/(pi-0)int_0^picos^4(x)sin(x)\ dx`

To solve this integral, I will introduce a u-substitution with `u=cos(x)`. The derivative of `u` is `-sin(x)`, so we divide through by that to integrate with respect to `u`. We also have to adjust the limits of integration by plugging them into `u=cos(x)`:
`1/piint_1^-1(u^4cancelsin(x))/-cancelsin(x)\ du=-1/piint_1^-1u^4\ du=`

`=-1/pi[u^5/5]_1^-1=-1/pi((-1)^5/5-1^5/5)=-1/pi(-1/5-1/5)=`

`=-1/pi(-2/5)=2/(5pi)`