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JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Area Under Curves

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 540 A is equal to ____.

Answer: 25

Solution

The region lies above both

y = x^{2}

and

y = (1 - x)^{2}

and below

y = 2 x (1 - x)

, and is symmetric about

x = \frac{1}{2}

. The boundaries meet at

x = \frac{1}{3}

, where

(1 - x)^{2} = 2 x (1 - x)

, and at

x = \frac{1}{2}

, where

x^{2} = (1 - x)^{2}

. Hence

A = 2 \int_{1/3}^{1/2} [2 x (1 - x) - (1 - x)^{2}] d x = 2 \int_{1/3}^{1/2} (4 x - 3 x^{2} - 1) d x = 2 [2 x^{2} - x^{3} - x]_{1/3}^{1/2} = 2 \cdot \frac{5}{216} = \frac{5}{108}

. Therefore

540 A = 540 \cdot \frac{5}{108} = 25

.

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Integer answer

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