Find the average value of the function f(x) = 4 {x}^(3)on the interval 1 \leqslant x \leqslant 4

Question

Find the average value of the function
`f(x) = 4 {x}^(3)`on the interval
`1 \leqslant x \leqslant 4`

Answer 1

Given the function f(x) defined as:

`f(x)=4x³`

The average value on the interval [1, 4] is given by:

`\bar{f}=(f(4)-f(1))/(4-1)`

Evaluating the function at x = 4 and x = 1:

`\begin{gathered} f(4)=4\cdot4³=256 \\ \\ f(1)=4\cdot1³=4 \end{gathered}`

Using these values for the average value of f:

`\begin{gathered} \bar{f}=(256-4)/(3)=(252)/(3) \\ \\ \therefore\bar{f}=84 \end{gathered}`