009B Sample Midterm 1, Problem 2

Find the average value of the function on the given interval.

$f(x)=2x^{3}(1+x^{2})^{4},~~~[0,2]$

Foundations:
The average value of a function $f(x)$ on an interval $[a,b]$ is given by $f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)dx$.

Solution:

Step 1:
Using the formula given in the Foundations sections, we have:
$f_{\text{avg}}={\frac {1}{2-0}}\int _{0}^{2}2x^{3}(1+x^{2})^{4}dx=\int _{0}^{2}x^{3}(1+x^{2})^{4}dx$

Step 2:

Final Answer:

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