2022 EJC J2 MYE P2 Q1

It is given that the origin $O$ lies on the curve $C$, which has equation $y=\text{f}\left( x \right)$. 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 $\int_{0}^{\alpha }{{{\left[ \text{f}\left( x \right) \right]}^{2}}\text{d}x}=\alpha {{\left[ \text{f}\left( \alpha \right) \right]}^{2}}$.

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