Find the mass of the surface S having the given mass density. S is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ , $z \geq 0$ ; the density at a point P on S is equal to the distance between P and the xy-plane.
A) $9 \pi$
B) $27 \pi$
C) 9
D) $3 \pi$

Question

Quizplus · Accepted Answer

Using spherical coordinates, the density function is given by $f(ho,	heta,\phi)=ho\sin\phi$. Since we are integrating over a hemisphere, $\phi \in [0,\pi/2]$ and $	heta \in [0,2\pi]$. Thus, we have:

\begin{align*}
	ext{Mass}&=\iiint\limits_Sf(ho,	heta,\phi)\ dV \
&= \int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{3} ho\sin\phi\cdotho^2\sin\phi\ dho\ d\phi\ d	heta \
&=\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{3} ho^3\sin^2\phi\ dho\ d\phi\ d	heta \
&=\int_{0}^{2\pi}\int_{0}^{\pi/2} \left[ \frac{1}{4}ho^4\sin^2\phi ight]_0^3\ d\phi\ d	heta \
&= \int_{0}^{2\pi}\int_{0}^{\pi/2} \frac{27}{4} \sin^2\phi\ d\phi\ d	heta \
&=\int_{0}^{2\pi} \left[-\frac{27}{8}\cos2\phiight]_0^{\pi/2}\ d	heta \
&= \frac{27}{4} \pi
\end{align*}

Therefore, the answer is B.

Find the Mass of the Surface S Having the Given x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9

Find the Mass of the Surface S Having the Given $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$