Center of Mass of a Circle How would one find the center of mass of a circle? The center of mass of a rod is given by:
$$\frac{1}{M}\int^{L}_{0}\rho x dx$$
So, for a sphere, it would be an area integral, such as:
$$\frac{1}{M}\int^{?}_{?} \rho ? dA$$ 
The "?" mean I don't know what to put there. 
This is an area integral, which means it needs to be taken over the two axises, correct? 

When one knows, or can divine, the answer in advance there is often a co-ordinate system that makes the problem trivial. For the Center of Mass of a sphere, integrate from r = 0 to R, the two semi-spheres above and below the x-y plane. That then trivially becomes the integral from 0 to R of ( (2 * pi * r^2) - (2 * pi * r^2) ) * dr. Amazingly, this integral evaluates to the origin! 
