Question

Find the average value of the function f(x) = 4x ^ 3 on the interval 1 <= x <= 3

Answer 1

Answer: We have to find the average value of the function:

`f(x)=4x^3`

The average value of a function is defined as follows:

`f_(avg)=(1)/(b-a)\int_a^bf(x)dx\Rightarrow(a,b)\text{ is interval }`

By using the definition of the average value of the function, the average is determined as follows:

`\begin{gathered} \begin{equation*} f_(avg)=(1)/(b-a)\int_a^bf(x)dx \end{equation*} \\ \\ b=3,a=1 \\ -------------------------- \\ \therefore\rightarrow \\ \\ f_(avg)=(1)/(3-1)\int_1^34x^3dx=(1)/(2)\int_1^34x^3dx \\ \\ \\ f_(avg)=(4)/(2)\int_1^3x^3dx=2\int_1^3x^3dx \\ \\ \\ f_(avg)=2\int_1^3x^3dx=2[(x^4)/(4)\Rightarrow(1,3)] \\ \\ \\ f_(avg)=2[((3)^4)/(4)-((1)^4)/(4)]=2[(81)/(4)-(4)/(4)]=2[(77)/(4)] \\ \\ --------------------- \\ f_(avg)=38.5 \end{gathered}`

Therefore the answer is 38.5.