Center of mass in a hemispherical tank with varying density

[vintwc]

Homework Statement

Let Ω be a tank whose shape is that of the lower hemisphere of radius R. The tank with a muddy suspension whose density ρ is ρ(x,y,z):=e^-h(x,y,z), where h(x,y,z) is the height of (x,y,z) above the lowest point of the tank. Find the center of mass in the tank

Homework Equations

The Attempt at a Solution

First of all, how does one determine the height, h(x,y,z)? I guess it would be R but I am not able to give a reasoning to my guess. I would appreciate if someone could give me a graphical illustration on how to find the limits of integration for this problem as well (ignore this if it will cause too much hassle). Thanks

 

[willem2]

If z is the vertical coordinate, and z =0 at the bottom of the tank, then h(x,y,z) = z

 

[vintwc]

I got a feeling its z+R. Could anyone let me know what are the phi limits of integration?