2024 HCI P1 Q3

The region bounded by the curve with equation \(x=\frac{y}{\sqrt{2y-{{y}^{2}}}}\), the lines \(y=1\), \(y=1.6\) and the \(y\)-axis is rotated through \(2\pi \) radians about the \(y\)-axis to form a solid ornament.

1. Find the exact volume \({{V}_{1}}\) of the ornament, giving your answer in terms of \(\pi \).[4]
2. An ornament designer designs a different ornament by rotating the region bounded by another curve with equation \(x=\frac{by}{\sqrt{2y-{{y}^{2}}}}\), where \(b>0\), the lines \(y=1\), \(y=1.6\) and the \(y\)-axis. The region is rotated through \(2\pi \) radians about the \(y\)-axis. The volume generated is now \({{V}_{2}}\). State the ratio of \({{V}_{1}}\) to \({{V}_{2}}\).[1]