2023 DHS Promo Q10

The curves \({{C}_{1}}\) and \({{C}_{2}}\) have equations \(y=\sqrt{15x+4}\) and \(y={{x}^{2}}-2\) respectively. The region \(R\) is bounded by the \(y\)-axis, \({{C}_{1}}\) and \({{C}_{2}}\).

1. Show algebraically that the point with coordinates \(\left( 3,7 \right)\) is the only point of intersection between \({{C}_{1}}\) and \({{C}_{2}}\).[3]
2. Without the use of a graphing calculator, find the exact area of \(R\).[4]
3. Find the volume of revolution when \(R\) is rotated \(2\pi \) radians about the \(y\)-axis.[3]