Answer
$$\dfrac{3\pi}{10}$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$\int^{2\pi}_0 \int^3_0 \int^{z/3}_0 r \space dr \space dz d\theta =\int^{2\pi}_0 \int^3_0 \dfrac{z^4}{324}dz d\theta \\ =\int^{2\pi}_0 \int^3_0 \dfrac{1}{324} (z^4) \space dz d\theta \\=\int^{2\pi}_0 \dfrac{4}{20}d\theta \\=\dfrac{3\pi}{10}$$
