Find the mass of the surface $S$ having the given mass density.
$S$ is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 , z \geq 0$; the density at a point $P$ on $S$ is equal to the distance between $P$ and the $x y$-plane. Select the correct answer.
A) $64 \pi$
B) $512 \pi$
C) 64
D) $8 \pi$

Question

Quizplus · Accepted Answer

We can use a triple integral to find the total mass of the hemisphere. Let $f(x,y,z)$ be the density at the point $(x,y,z)$. Then, the total mass is given by:

$$
\int\int\int_{S}f(x,y,z)\,dV
$$

We are given that $f(x,y,z) = \sqrt{x^2 + y^2}$ at every point $(x,y,z)$ on the hemisphere. In spherical coordinates, the equation of the hemisphere becomes $ho = 4$, $	heta \in [0,2\pi]$, and $\phi \in [0,\pi/2]$. Thus, we have:

$$
\begin{aligned}
\int\int\int_{S}f(x,y,z)\,dV &= \int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{4}(ho\sin\phi)\sqrt{ho^2\sin^2\phi\cos^2	heta + ho^2\sin^2\phi\sin^2	heta}\,ho^2\sin\phi dho d\phi d	heta \
&= \int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{4}ho^3\sin^2\phi\sqrt{\cos^2	heta + \sin^2	heta}\,dho d\phi d	heta \
&= \int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{4}ho^3\sin^2\phi\,dho d\phi d	heta \
&= \frac{64\pi}{3}\int_{0}^{\pi/2}\sin^2\phi\,d\phi \
&= \frac{32\pi}{3}\int_{0}^{1}(1-u^2)\,du \
&= \frac{32\pi}{3} \left(u - \frac{u^3}{3}ight)\bigg\vert_{0}^{1} \
&= \frac{32\pi}{3} \cdot \frac{2}{3} \
&= \frac{64\pi}{3}
\end{aligned}
$$

Therefore, the mass of the hemisphere is $\frac{64\pi}{3}$. Since the density function is proportional to the distance from a point on $S$ to the $xy$-plane, we would expect the mass to be proportional to the total volume of the hemisphere, which is $\frac{2}{3}$ of the volume of the full sphere with radius $4$. The full sphere has volume $\frac{4}{3}\pi(4)^3 = \frac{256}{3}\pi$, so $\frac{2}{3}$ of this volume is $\frac{128}{3}\pi$. The ratio of the actual mass of the hemisphere to $\frac{128}{3}\pi$ is:

$$
\frac{\frac{64\pi}{3}}{\frac{128}{3}\pi} = \frac{1}{2}
$$

Therefore, the correct answer is $	extbf{(B)}$ $512\pi$.

Find the Mass of the Surface SSS Having the Given Mass Density

Find the Mass of the Surface $S$ Having the Given Mass Density