Question

What is the average value of the function `f(x)=e^4x^2` on the interval `[-1/4,1/4]`?

Answer 1

`e^4/48`

Explanation:

The average value of the function `f(x)` on the interval `[a,b]` is equal to:

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

Thus here the average value is

`1/(1/4-(-1/4))int_(-1/4)^(1/4)e^4x^2dx=e^4/(1/2)int_(-1/4)^(1/4)x^2dx=2e^4[x^3/3]_(-1/4)^(1/4)`

`=2e^4[(1/4)^3/3-(-1/4)^3/3]=2e^4[1/192-(-1/192)]=e^4/48`