$$y = {x^2}$$ and $$y = 2x(1 - x)$$

$$ \Rightarrow {x^2} = 2x - 2{x^2}$$

$$ \Rightarrow x = 0,x = {2 \over 3}$$

Now,

$$y = {(1 - x)^2}$$ and $$y = 2x(1 - x)$$

$$ \Rightarrow 1 + {x^2} - 2x = 2x - 2{x^2}$$

$$ \Rightarrow 3{x^2} - 4x + 1 = 0$$

$$x = 1,x = {1 \over 3}$$

$$\therefore$$ $$A = \int\limits_{{1 \over 3}}^{{2 \over 3}} {(2x - 2{x^2})dx - \left\{ {\int\limits_{{1 \over 3}}^{{1 \over 2}} {{{(1 - x)}^2}dx + \int\limits_{{1 \over 2}}^{{2 \over 3}} {{x^2}dx} } } \right\}} $$

= $$\left( {{x^2} - {{2{x^3}} \over 3}} \right)_{{1 \over 3}}^{{2 \over 3}} - \left\{ {\left( {{{{{\left( {x - 1} \right)}^3}} \over 3}} \right)_{{1 \over 3}}^{{1 \over 2}} + \left( {{{{x^3}} \over 3}} \right)_{{1 \over 2}}^{{2 \over 3}}} \right\}$$

$$ = {5 \over {108}}$$

$$\therefore$$ $$540A = 25$$