Question

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

Answer 1

`(1-e^(-25))/10`

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/(5-0)int_0^5xe^(-x^2)\ dx`

To work out the integral, I will introduce a u-substiution with `u=-x^2`. The derivative of `u` is then `-2x`, so we divide by that to integrate with respect to `u`. We also have to adjust the limits of integration according to our substitution, so we plug the limits into `u=-x^2`:

`-0^2=0`

`-5^2=-25`

This results in:
`1/5int_0^(-25)(cancelxe^u)/(-2cancel(x))\ du=-1/10int_0^(-25)e^u\ du=-1/10[e^u]_0^(-25)=`

`=-1/10(e^(-25)-e^0)=(1-e^(-25))/10`