Find the average value of 𝑓(𝑥)=𝑥^3 on the interval [1,4] ?

Answer 1

Take the integral from 1 to 4 to get the area under the curve, then divide by (4-1) or 3:

Integral = ∫ x³ dx = (1/4) x⁴ +> Area = { (1/4)*4⁴ - (1/4)*1⁴ } / 3 = 63.75 / 3 = 21.25

Answer 2

the average of a function is defined as 1/(Integral Range) times the integral over the range.

In this case, 1(4-1)Integral of x^3 over x=1 to 4.

That becomes (1/3)(1/4)x^4 from x=1 to 4. This becomes (1/12)(4^4 - 1^4) = (1/12)(256-1) =21.25