In order to find the centre of mass of the hemisphere consider a cylindrical element of thickness $\delta x$ at distance $x$ from the centre of the base $O$. Then the radius of this slice is $\sqrt{(}{a}^2-x^2)$. The position of the centre of mass is found by taking moments about $O$ and summing these moments for all the 'slices' by integrating:

${\displaystyle \frac{2}{3}w\pi a^3\overline{x}={\int }_0^{a}x\times w\pi (a^2-x^2)\mathrm{dx}}$

which gives

${\displaystyle \overline{x}=\frac{3a}{8}.}$

Similarly to find the centre of mass of the uniform cone, a distance $\overline{x}$ along the axis of symmetry from the vertex $V$, consider cylindrical elements of thickness $\delta x$ and radius $\frac{\mathrm{ax}}{h}$ (from similar triangles). The position of the centre of mass is again found by taking moments, in this case about $V$:

${\displaystyle \frac{1}{3}w\pi a^{2}h\overline{x}={\int }_0^{h}x\times w\pi \frac{a^2{x}^2}{h^2}\mathrm{dx}}$

which gives

${\displaystyle \overline{x}=\frac{3h}{4}.}$