Let
$D = \{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2\le z^2, 0\le z \le {3}\}.$
How do I integrate
$\displaystyle \iiint\limits_{D} \left({\frac{y^2+x^2}{27}}\right) \,\mathrm{d}V$
I know that in a cylindrical coordinate system $(x,y,z)\to(r\cos\theta,r\sin\theta,z) $ and we have $r^2 = x^2+y^2$.
I have hard time to understand the limit values of the integral.
Since $D$ is the region $r\in [0,\,z],\,\theta\in[0,\,2\pi],\,z\in[0,\,3]$ with $dV=rdrd\theta dz$, your integral is $$\int_0^{2\pi}d\theta\int_0^3dz\left[\int_0^z\frac{r^3}{27}dr\right]=\int_0^{2\pi}d\theta\int_0^3\frac{z^4}{108}dz=2\pi\frac{3^5}{540}=\frac{9\pi}{10}.$$