Question

What is the average value of the function `f(x) = 2x^3(1+x^2)^4` on the interval `[0,2]`?

Answer 1

The average value is `4948/5 = 989.6`

Explanation:

The average value of `f` on interval `[a,b]` is `1/(b-a) int_a^b f(x) dx`

So we get:

`1/(2-0)int_0^2 2x^3 (x^2+1)^4 dx = 2/2 int_0^2 x^3(x^8+4x^6+10x^4+4x^2+1) dx`

`= int_0^2 (x^11+4x^9+10x^7+4x^5+x^3)dx`

`= x^12/12+(4x^10)/10 + (6x^8)/8+(4x^6)/6+x^4/4]_0^2`

`= (2)^12/12+(2(2)^10)/5 + (3(2)^8)/4+(2(2)^6)/3+(2)^4/4`

`= 4948/5 = 9896/10=989.6`