The region in the first quadrant enclosed by the curve \(y = \dfrac{b(a^2 - x^2)^{1/2}}{a}\) and the lines \(y = 0\), \(x = 0\), \(x = ka\) \((0 < k \le 1)\), is rotated through four right angles about the \(x\)-axis to form a solid of revolution. Show that the volume of the solid is \(\tfrac{1}{3} \pi ab^2k(3-k^2)\).
If this solid is of uniform density find the coordinates of its centre of mass.
By considering the case \(b = a\), \(k = 1\), show that the centre of mass of a uniform solid hemisphere of radius \(a\) is at a distance \(\tfrac{3}{8}a\) from the centre.
