Chapter 15 - Multiple Integrals - 15.7 Triple-Integrals in Cylindrical Coordinates - 15.7 Exercises - Page 1083: 18

$\dfrac{64\pi}{3}$

Consider $I=\iiint_EZdV=\int_0^{2\pi} \int_{0}^{2}\int_{r^2}^{4} z \times r dr dz d\theta$ $=\int_0^{2\pi} \int_{0}^{2}[\dfrac{z^2}{2}]_{r^2}^{4} \times ( r) dr dz d\theta$ $=\int_0^{2\pi} \int_{0}^{2}[ (8r-\dfrac{r^5}{2}) dr ]dz d\theta$ $=\int_0^{2\pi}[\dfrac{8r^2}{2}-\dfrac{r^6}{12}]_{0}^{2} d\theta$ $=\dfrac{64\pi}{3}$