Answer
$\dfrac{64 \pi^3}{27}$
Work Step by Step
Re-arrange the integral as follows:
$\int_{0}^{\frac{4 \pi}{3}} \int_{0}^{2 \pi} r dr d \phi= \int_{0}^{2 \pi}
\dfrac{r^2}{2}]_{0}^{\frac{4 \pi}{3}} d \phi$
This implies that
$\dfrac{8}{9} \int_{0}^{2 \pi} \phi^2 d\phi=(\dfrac{8}{9})[ \dfrac{\phi^3}{3}]_{0}^{2 \pi} $
Thus, we have
$(\dfrac{8}{9})[ \dfrac{\phi^3}{3}]_{0}^{2 \pi} =\dfrac{64 \pi^3}{27}$
