The area of the region { ( x , y ) : x 2 y | x 2 - 4 | , y 1 } is [2023]

Question

The area of the region ${(x, y) : x^{2} \leq y \leq | x^{2} - 4 |, y \geq 1}$ is [2023]

Smart Achievers · Accepted Answer

(4)

$Required area = 2 (\int_{1}^{2} \sqrt{y} d y + \int_{2}^{4} \sqrt{4 - y} d y)$

$= 2 ({[\frac{y^{3 / 2}}{3 / 2}]}_{1}^{2} - {[\frac{{(4 - y)}^{3 / 2}}{3 / 2}]}_{2}^{4})$

$= \frac{4}{3} [(2 \sqrt{2} - 1) - (0 - 2 \sqrt{2})] = \frac{4}{3} (4 \sqrt{2} - 1)$

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