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

$ A =2 \int\limits_{\frac{1}{3}}^{\frac{1}{2}}\left(2 x-2 x ^2-(1- x )^2\right) dx $
$ =2\left[2 x ^2- x ^3- x \right]_{1 / 3}^{1 / 2} $
$ \therefore A =\frac{5}{108} \Rightarrow 540 A =\frac{5}{108} \times 540=25$