I have been calculating the center of masses of various objects using triple integrals, however, one thing I am struggling is calculating the center of mass of hollow objects. My first approach was to go in a similar fashion as we do for solid objects, however, setting the bounds differently.
For example, for a solid hemisphere with uniform density, the center of mass would be $$\frac{\frac{M}{\frac43 \pi R ^3}\int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{R} \rho ^2 \sin \phi \cos \phi d\rho d\phi d\theta}{M}$$ For the c.o.m. of a hollow hemisphere with a similar uniform density, I was thinking of evaluating a similar integral: $$\frac{\frac{M}{4 \pi R ^2}\int_{0}^{2 \pi} \int_{0}^{\frac{\pi}{2}} \int_{R}^{R + dR} \rho ^2 \sin \phi \cos \phi d\rho d\phi d\theta}{M}$$
where $dR$ indicates the change in radius. I quickly realised that this approach is flawed. What correction should I make to this approach to find the c.o.m. of a hollow body?