Chapter 14.6, Problem 69E

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.

69.    The cylinder $x^2+y^2=a^2,0\le z\le 2$, with density $\rho \left(x,y,z\right)=1+z$