Chapter 14.6, Problem 68E

Mass and center of mass Let S be a surface that represents a thin shell with density ρ. The moments about the coordinate planes (see Section 13.6) are $M_{yz}={\displaystyle {\iint }_{S}x\rho \left(x,y,z\right)dS,M_{xz}=}{\displaystyle {\iint }_{S}y\rho \left(x,y,z\right)dS}$, and $M_{xy}={\displaystyle {\iint }_{S}z\rho \left(x,y,z\right)dS}$. The coordinates of the center of mass of the shell are $\bar{x}=\frac{M_{yz}}{m},\bar{y}=\frac{M_{xz}}{m},\bar{z}=\frac{M_{xy}}{m}$, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible.

68.    The constant-density half cylinder

$x^2+z^2=a^2,-h/2\le y\le h/2,z\ge 0$