Question

If the x-axis divide the area of the region bounded by the parabolas $y=4x-x^2$ and $y=x^2-x$ in the ratio of $a:b$ then $ab$ is equal to

Answer 1

Roots of $x^2-x=0$ are 0,1. and that of $4x-x^2$ are 0,4.
intersection of both parabola : 0, $\frac{5}{2}$
So,
$\frac{{\int }_0^1(4x-x^2)dx+{\int }_1^{\frac{5}{2}}\left[\right(4x-x^2)-(x^2-x\left)\right]dx}{{\int }_0^1-(x^2-x)dx}=\frac{a}{b}$
$\frac{a}{b}=\frac{\frac{5}{3}+\frac{27}{8}}{\frac{1}{6}}=\frac{121}{4}$
$ab=121\times 4=484$