Answer
$\displaystyle \frac{2kR^{2}}{3}$
Work Step by Step
v is a function of r, [a,b]=[0,R]
R and k are constants.
Average value $=\displaystyle \frac{1}{b-a}\int_{a}^{b}v(r)dr$
$= \displaystyle \frac{1}{R-0}\int_{0}^{R}k(R^{2}-r^{2})dr$
$=\displaystyle \frac{k}{R}[R^{2}r-\frac{r^{3}}{3}]_{0}^{R}$
$=\displaystyle \frac{2kR^{2}}{3}$
