Question

Let $A$ be the area of the region
$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$. 
Then $540 A$ is equal to ______.

Answer 1

Correct Answer: 25

$A=2\underset{\frac{1}{3}}{\overset{\frac{1}{2}}{\int }}\left(2x-2x^2-(1-x{)}^2\right)dx$
$=2{\left[2x^2-x^3-x\right]}_{1/3}^{1/2}$
$\therefore A=\frac{5}{108}\Rightarrow 540A=\frac{5}{108}\times 540=25$
So , the correct answer is 25.