2022 EJC J2 MYE P2 Q1

It is given that the origin \(O\) lies on the curve \(C\), which has equation \(y=\mathrm{f}\left( x \right)\). The region \({{R}_{1}}\) is bounded by the curve \(C\), the horizontal line \(y=k\) and the \(y\)-axis, while the region \({{R}_{2}}\) is bounded by the curve \(C\) and the horizontal line \(y=k\). The line \(y=k\) intersects the curve \(C\) at two points, one of which has \(\alpha \) as its \(x\)-coordinate (see diagram). It is given that the volume of the solid obtained by rotating \({{R}_{1}}\) about the \(x\)-axis is equal to the volume of the solid obtained by rotating \({{R}_{2}}\) about the \(x\)-axis. Show that \(\alpha \) satisfies the equation \(\displaystyle\displaystyle \displaystyle\int_{0}^{\alpha }{{{\left[ \mathrm{f}\left( x \right) \right]}^{2}}\mathrm{d}x}=\alpha {{\left[ \mathrm{f}\left( \alpha \right) \right]}^{2}}\).[4]