Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 15 / c / ii / Solution

For positive constant mass density ${\rho }_0$, reflection symmetry makes the horizontal coordinates of the centre of mass zero. The total mass and the vertical first moment areTheir ratio givesThis is the centre of the liquid volume, rather than of the curved wall. The density cancels because it is uniform; the larger disks near the top place the centre above half-height.