Let A be the area of the region { ( x , y ) : y x 2 , y ( 1 - x ) 2 , y 2 x ( 1 - x ) } . Then 540A is equal to ___________ . [2023]

Question

Let A be the area of the region ${(x, y) : y \geq x^{2}, y \geq {(1 - x)}^{2}, y \leq 2 x (1 - x)}$ . Then 540A is equal to ___________ . [2023]

Smart Achievers · Accepted Answer

(25)

Region ${(x, y) : y \geq x^{2}, y \geq {(1 - x)}^{2}, y \leq 2 x (1 - x)}$

$y = x^{2}$ $\dots (i)$

$y = {(1 - x)}^{2}$ $\dots (i i)$

$y = 2 x (1 - x)$ $\dots (i i i)$

Solving (ii) and (iii), we get

$x = \frac{1}{3} and x = 1$

$∴$ $Required area A = 2 \int_{1 / 3}^{1 / 2} [2 x - 2 x^{2} - {(1 - x)}^{2}] d x$

$= 2 \int_{1 / 3}^{1 / 2} (2 x - 2 x^{2} - 1 - x^{2} + 2 x) d x$

$= 2 \int_{1 / 3}^{1 / 2} (- 3 x^{2} + 4 x - 1) d x$ $= {[- x^{3} + 2 x^{2} - x]}_{1 / 3}^{1 / 2} = \frac{5}{108}$

$∴$ $540 A = 540 \times \frac{5}{108} = 25$

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