Answer
$\dfrac{1}{10}\pi kh^5$
Work Step by Step
Consider $I_z=\iiint_{E} (x^2+y^2) \rho(x,y,z) dV$
or, $=k\int_{0}^{2 \pi} \int_{0}^{h}\int_{0}^{z}(r^2)r dr dz d\theta $
or, $=k\int_{0}^{2 \pi} \int_{0}^{h}\int_{0}^{z}(r^3) dr dz d\theta $
or, $=k \int_{0}^{2 \pi} d\theta \int_{0}^{h} \int_{0}^{z}(r^3) dr dz $
or, $=2\pi k \int_0^h [\dfrac{z^4}{4}] dz $
or, $=2\pi k [\dfrac{z^5}{20}]_0^h $
or, $I_z=\dfrac{1}{10}\pi kh^5$
