A uniform solid right circular cylinder of radius \(a\) and height \(2a\) stands on a horizontal plane. Rectangular co-ordinate axes \(Ox\), \(Oy\) are taken in the plane of the base, with \(O\) at the centre of the base. The cylinder is cut into two equal parts by the plane which passes through the line \(x = -a\) and makes an angle of \(45^\circ\) with the horizontal. Justify the formula \[\begin{equation*} \int_{-a}^a 2(x+a) \sqrt{a^2 - x^2} \:dx \end{equation*}\]

for the volume of the lower half, and verify by calculation of the integral that the volume is \(\pi a^3\).

Prove that the centre of mass of this lower half is vertically above the point \((\tfrac{1}{4}a ,0)\).