Answer
$\dfrac{162\pi}{5}$
Work Step by Step
Conversion of rectangular to cylindrical coordinate system gives:
$r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Here, $x=r \cos \theta; y=r \sin \theta, z=z$
As the per problem, $z=9-x^2-y^2 \implies z=9-r^2$
In the cylindrical coordinate system, we have $I=\int_0^{\pi} \int_{0}^{3}\int_{0}^{9-r^2} (r \cdot r) dz dr d\theta$
or, $I=\int_0^{\pi} \int_{0}^{3} [r^2z]_{0}^{9-r^2}dr d\theta$
or, $I=\int_0^{\pi} \int_{0}^{3} (9r^2-r^4)dr d\theta$
or, $I=\int_0^{\pi} [3r^3-\dfrac{r^5}{5}]_{0}^{3} d\theta$
or, $I=[\dfrac{162\theta}{5}]_0^{\pi}$
or, $I=\dfrac{162\pi}{5}$
