Surface integrals: Finding the center of mass of a thin sheet with the shape of surface S

Question

How do I find the center of mass of a thin sheet when $S$ is the upper hemisphere $x^2+y^2+z^2=a^2$ with $z\ge 0$ and density $\delta(x,y,z)= k$ (constant).

Also I have to compute this using surface integrals...I would have been able to figure it out if it didn't specify that...

Answer 1

Hint: The coordinates $(x_{c}, y_{c}, z_{c})$ of the center of mass are given by

$x_{c} = \iint x \,dxdydz$,

$y_{c} = \iint y \,dxdydz$,

$z_{c} = \iint z \,dxdydz$.

Use Green's theorem to turn these into surface integrals.

Answer 2

when

$$ dV= dx\,dy\,dz,\quad V= \iiint dx\,dy\,dz\,,$$

$$x_{c} = \iiint x \,dV/ V; \,y_{c} = \iiint y \,dV/ V ; \,z_{c} = \iiint z\,dV/ V \, $$

Answer 3

Mass$=\iiint_{\displaystyle s} \delta dV$

First Moments About Coordinate Planes:

$M_{yz}=\iiint_{s}x\;dV,\;\;\;\; M_{xz}=\iiint_{s}y\;dV, \;\;\;\;\;\;M_{xy}=\iiint_{s}z\;dV$

Center Of Mass:= $(x',y',z')=\left(\frac{M_{\displaystyle yz}}{M},\frac{M_{\displaystyle xz}}{M},\frac{M_{\displaystyle xy}}{M}\right)$