Question

What is the average value of the function `F(x) = xe ^ (x^2)` on the interval `[0,2]`?

Answer 1

`1/4 e^{4}-1/4 approx 13.3995`

Explanation:

The average value of a continuous function `F` over an interval `[a,b]` is `1/(b-a) int_{a}^[b}\ F(x)\ dx`.

For `F(x)=x e^{x^{2}}` over the interval `[0,2]`, this becomes

`1/2 int_{0}^{2}x e^{x^{2}}\ dx`

The integral can be done using the substitution `u=x^{2}`, making `du=2x\ dx`. Also, when `x=0`, `u=0` and when `x=2`, `u=2^{2}=4`. Therefore, the quantity above can be written as

`1/2 int_{0}^{4} 1/2 e^{u}\ du=[1/4 e^{u}]_{0}^{4}=1/4 e^{4}-1/4 e^{0}=1/4 e^{4}-1/4 approx 13.3995`