Part $1$: Mostly classical calculus

Problem $1.{?}^{(****)}$ It builds character in $u.$ The density of a "squeezed" cone is equal to its height at any point. What is the average density of the cone between any two heights? The solution for this problem is given as

$\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{}}}x*(\frac{x^2}{3}+2{)}^{2}dx=\frac{((b^2+6{)}^3-(a^2+6{)}^3)}{54(b-a)}$

Verify that this is correct using integration, and by finding the derivative of the indefinite integral version below:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x*(\frac{x^2}{3}+2{)}^{2}dx=\frac{}{2}(\frac{x^2}{3}+2{)}^3+C$