Consider the solid that is bounded below by the cone z = sqrt{3x^{2}+3y^{2}} and above by the sphere x^{2} +y^{2} + z^{2} = 16..Set up only the appropriate triple integrals in cylindrical and spherical coordinates needed to find the volume of the solid.

Step 1
To set the triple integral in cylindrical coordinates
By using relation,
$x=r\cos \theta$
$y=r\sin \theta$
$z=z$
Thus,
The cone $z=\sqrt{3x^2+3y^2}$ in cylindrical coordinate becomes,
$z=\sqrt{3r^2}=\sqrt{3r}$
And the sphere become $r^2+z^2=16$
To find the limit of r,
Consider,
$\Rightarrow \sqrt{3r^2}=\sqrt{16-r^2}$
$\Rightarrow 3r^2=16-r^2$

$\Rightarrow r^2=4$
$\Rightarrow r=2$
Step 2
Thus, we can describe the region as
$E=\{(r,\theta ,z)|0\le \theta \le 2\pi |0\le r\le 2\sqrt{3r}\le z\le \sqrt{16-r^2}\}$
Hence, the triple integral for the volume by cylindrical coordinates is
$V={\int }_0^{2\pi }{\int }_0^2{\int }_{\sqrt{3r}}^{\sqrt{16-r^2}}rdzdrd\theta$
Step 3
Now, to set the triple integral in spherical coordinates
Since,
$p^2=x^2+y^2+z^2$
$\tan \theta =\frac{y}{x}$
$\phi =\arccos (\frac{z}{\sqrt{x^2+y^2+z^2}})$
The sphere $x^2+y^2+z^2=16gives{p}^2=16=>p=4$
From the cone
$z=\sqrt{3x^2+3y^2}=\sqrt{3r}$
$\Rightarrow p\cos (\phi )=\sqrt{3}p\sin (\phi )$
$\Rightarrow p\tan (\phi )=\frac{1}{\sqrt{3}}$
$\phi =\frac{\pi }{6}$
Step 4
Thus, we can describe the region as
$E=\{(p,\phi ,\theta )|0\le p\le 4,0\le \phi \le \frac{\pi }{6},0\le \theta \le 2\pi \}$
Hence, the triple integral for the volume of the solid by spherical coordinate is
$V={\int }_0^{2\pi }{\int }_0^{\frac{\pi }{6}}{\int }_0^4{p}^2\sin (\phi )dpd\phi d\theta$
Step 5
Now, evaluating the integral of cylindrical coordinate we get.
$\Rightarrow V={\int }_0^{2\pi }{\int }_0^2{\int }_{\sqrt{3r}}^{\sqrt{16-r^2}}rdzdrd\theta$
$\Rightarrow V=17.9582$
And evaluating the integral of spherical coordinate
$\Rightarrow V={\int }_0^{2\pi }{\int }_0^{\frac{\pi }{6}}{\int }_0^{\sqrt{4}}{r}^2\sin (\phi )dpd\phi d\theta$
$\Rightarrow V=17.9582$
Thus, by both coordinate systems, we get the same volume.
Therefore, both triple integrals are appropriate.