Question

Let $P\left(q\right)=\left\{\right(x,y):y^2\le 4x,0\le x\le q$
and $A\left(q\right)$ is the area of the region $P\left(q\right).$
If for a value $\alpha (0<\alpha <3),A\left(\alpha \right):A\left(3\right)=1:3,$ then the value of ${\alpha }^3$ =

Answer 1

$A\left(q\right)=2\times {\int }_0^{q}2\sqrt{x}dx=\frac{8}{3}{q}^{3/2}\frac{A\left(\alpha \right)}{A\left(3\right)}=\frac{1}{3}=\frac{{\alpha }^{3/2}}{3^{3/2}}\Rightarrow {\alpha }^3=3$