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

\(E = \{(r, \theta, z)|0 \leq \theta \leq 2 \pi|0 \leq r \leq 2 \sqrt{3r} \leq z \leq \sqrt{16-r^{2}}\}\)

Hence, the triple integral for the volume by cylindrical coordinates is

\(V=\int_{0}^{2 \pi} \int_{0}^{2} \int_{\sqrt{3r}}^{\sqrt{16-r^{2}}} r d z d r d \theta\)

Step 3

Now, to set the triple integral in spherical coordinates

Since,

\(p^{2}=x^{2}+y^{2}+z^{2}\)

\(\tan \theta = \frac{y}{x}\)

\(\varphi= \arccos (\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}})\)

The sphere \(x^2 + y^2 + z^2 = 16\ gives\ p^2 = 16 => p = 4\)

From the cone

\(z=\sqrt{3x^{2}+3y^{2}}=\sqrt{3r}\)

\(\Rightarrow p \cos (\varphi)=\sqrt{3}p \sin (\varphi)\)

\(\Rightarrow p \tan (\varphi)=\frac{1}{\sqrt{3}}\)

\(\varphi = \frac{\pi}{6}\)

Step 4

Thus, we can describe the region as

\(E = \{(p, \varphi, \theta)|0 \leq p \leq 4, 0 \leq \varphi \leq \frac{\pi}{6}, 0 \leq \theta \leq 2 \pi\}\)

Hence, the triple integral for the volume of the solid by spherical coordinate is

\(V=\int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{6}} \int_{0}^{4} p^{2} \sin (\varphi)\ d\ pd\ \varphi\ d\ \theta\)

Step 5

Now, evaluating the integral of cylindrical coordinate we get.

\(\Rightarrow V = \int_{0}^{2 \pi} \int_{0}^{2} \int_{\sqrt{3 r}}^{\sqrt{16-r^{2}}} rdzdrd \theta\)

\(\Rightarrow V = 17.9582\)

And evaluating the integral of spherical coordinate

\(\Rightarrow V = \int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{6}} \int_{0}^{\sqrt{4}} r^{2} \sin (\varphi) dpd \varphi d \theta\)

\(\Rightarrow V = 17.9582\)

Thus, by both coordinate systems, we get the same volume.

Therefore, both triple integrals are appropriate.