Question

It the area enclosed by the parabolas $P_1:2y=5x^2$ and $P_2:x^2-y+6=0$ is equal to the area enclosed by $P_1$ and $y=\alpha x,\alpha >0$, then ${\alpha }^3$ is equal to_____

Answer 1

Answer: 600

$\text{ Area }=2\stackrel{2}{\int }\left(x^2+6-\frac{5x^2}{2}\right)dx=\underset{0}{\overset{\frac{2\alpha }{5}}{\int }}\left(\alpha x-\frac{5x^2}{2}\right)dx$
$\Rightarrow \underset{0}{\overset{\frac{2\alpha }{5}}{\int }}\left(\alpha x-\frac{5x^2}{2}\right)dx=16$
$\Rightarrow {\alpha }^3=600$