Centre of mass
The problem statement, all variables and given/known data
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
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. Thanks
Alright following the my notes so far which had a sort of similar but different question, I guess that the height is z+R? Assuming if this is right, the limits of integration will be [0,R]x[-Pi/2,-0]x[0,2*Pi] (since we are looking at the lower hemisphere).
However, if I try to calculate the first moments, I get this horrible expression which I can't integrate. Maybe this indicates that I am moving in the wrong direction?