This solid has symmetry across each of the planes [tex]x=0[/tex], [tex]y=0[/tex], and [tex]z=0[/tex], which means the overall volume of the solid is 8 times the volume of the solid **contained in the first octant**.
In cylindrical coordinates, the region in the first octant is the set
[tex]R = \left\{(r,\theta,z) ~:~ 3\le r\le6\text{ and }0\le\theta\le\dfrac\pi2\text{ and }0\le z\le\sqrt{36-r^2}\right\}[/tex]
so the volume is
[tex]\displaystyle 8\iiint_RdV = 8 \int_0^{\pi/2} \int_3^6 \int_0^{\sqrt{36-r^2}} r\,dz\,dr\,d\theta \\\\ ~~~~~~~~ = 8 \int_0^{\pi/2} \int_3^6 r\sqrt{36-r^2} \, dr \, d\theta \\\\ ~~~~~~~~ = 4\pi \int_3^6 r \sqrt{36 - r^2} \, dr \\\\ ~~~~~~~~ = 4\pi \int_{27}^0 \sqrt s \left(-\frac12\,ds\right) \\\\ ~~~~~~~~ = 2\pi \int_0^{27} \sqrt s\,ds \\\\ ~~~~~~~~ = 2\pi\cdot\frac23 \left(27^{3/2} - 0^{3/2}\right) = \boxed{108\sqrt3\,\pi}[/tex]
