If 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 ____________.

Area =2 integral _0^2(x^2+6-(5 x^2 / 2)) dx =integral _0^(2 alpha / 5)(alpha x-(5 x^2 / 2)) d x => integral _0^(2 alpha / 5)(alpha x-(5 x^2 / 2)) d x=16 => alpha^3=600

If the area enclosed by the parabolas P12y5x2 and P2x2y... | (Mathematics)

Area $=2 \int_0^2\left(x^2+6-\frac{5 x^2}{2}\right) dx\\$ = $\int_0^{\frac{2 \alpha}{5}}\left(\alpha x-\frac{5 x^2}{2}\right) d x \\$

$\Rightarrow \int_0^{\frac{2 \alpha}{5}}\left(\alpha x-\frac{5 x^2}{2}\right) d x=16 \\$

$\Rightarrow \alpha^3=600$

Explanation

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