Further centres of mass

Centres of mass by calculus

Slice the body into thin strips or discs, treat each as a particle, and add with integrals: for a lamina under $y=f(x)$, $\bar x=\frac{\int xy\,dx}{\int y\,dx}$; for a solid of revolution, $\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}$. Standard results: cone $\frac h4$ from the base, hemisphere $\frac{3r}8$ from the flat face.