Let A be the area of the region \ ( x, y ) : y ≥ x ^ 2 , y ≥ ( 1 -...

Given,

$A$ be the area of the region

\left\{( x, y ) : y \ge {x}^{2}, y \ge ( 1 - x{)}^{2}, y \le 2x ( 1 - x )\right\}$</p><p>Now solving$y = {x}^{2} \text{ and } y = 2x\left(1 - x\right)$we get,$x = 0, \dfrac{2}{3}$</p><p>And solving$y = {\left(1 - x\right)}^{2} \text{ and } y = 2x\left(1 - x\right)$we get, </p><p>$⇒ 1 + {x}^{2} - 2x = 2x - 2{x}^{2}$</p><p>$⇒ 3{x}^{2} - 4x + 1 = 0$</p><p>$⇒ x = 1, \dfrac{1}{3}$</p><p>Now on plotting the diagram of the above region we get,</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/b5210be6-68c5-4a37-9ddf-7e9c63368b21-9896001.png" style="width: 500px; height: 437px;" /></p><p>Now from the above diagram area of the shaded region will be,</p><p>$A = {∫}_{\dfrac{1}{3}}^{\dfrac{2}{3}}\left(2x - 2{x}^{2}\right)dx - \left\{{∫}_{\dfrac{1}{3}}^{\dfrac{1}{2}}{\left(1 - x\right)}^{2}dx + {∫}_{\dfrac{1}{2}}^{\dfrac{2}{3}}{x}^{2}dx\right\}$</p><p>$⇒ A = {\left[{x}^{2} - \dfrac{2{x}^{3}}{3}\right]}_{\dfrac{1}{3}}^{\dfrac{2}{3}} - \left\{{\left[\dfrac{{\left(x - 1\right)}^{3}}{3}\right]}_{\dfrac{1}{3}}^{\dfrac{1}{2}} + {\left[\dfrac{{x}^{3}}{3}\right]}_{\dfrac{1}{2}}^{\dfrac{2}{3}}\right\}$</p><p>$⇒ A = \dfrac{5}{108}$</p><p>$∴ A = \dfrac{5}{108} ⇒ 540A = \dfrac{5}{108} \times 540 = 25$