Answer: We have to find the average value of the function:
[tex]f(x)=4x^3[/tex]
The average value of a function is defined as follows:
[tex]f_{avg}=\frac{1}{b-a}\int_a^bf(x)dx\Rightarrow(a,b)\text{ is interval }[/tex]
By using the definition of the average value of the function, the average is determined as follows:
[tex]\begin{gathered} \begin{equation*} f_{avg}=\frac{1}{b-a}\int_a^bf(x)dx \end{equation*} \\ \\ b=3,a=1 \\ -------------------------- \\ \therefore\rightarrow \\ \\ f_{avg}=\frac{1}{3-1}\int_1^34x^3dx=\frac{1}{2}\int_1^34x^3dx \\ \\ \\ f_{avg}=\frac{4}{2}\int_1^3x^3dx=2\int_1^3x^3dx \\ \\ \\ f_{avg}=2\int_1^3x^3dx=2[\frac{x^4}{4}\Rightarrow(1,3)] \\ \\ \\ f_{avg}=2[\frac{(3)^4}{4}-\frac{(1)^4}{4}]=2[\frac{81}{4}-\frac{4}{4}]=2[\frac{77}{4}] \\ \\ --------------------- \\ f_{avg}=38.5 \end{gathered}[/tex]
Therefore the answer is 38.5.
