Let the area of the region be A. Then 6A is equal to : [2025]

Question

Let the area of the region ${(x, y) : 2 y \leq x^{2} + 3, y + | x | \leq 3, y \geq | x - 1 |}$ be A. Then 6A is equal to : [2025]

Smart Achievers · Accepted Answer

(1)

Required area, A = Rectangle ABDE – Area of region EDC

$\Rightarrow A = 4 - 2 \int_{0}^{1} ((3 - x) - (\frac{x^{2} + 3}{2})) d x$

$\Rightarrow A = 4 - 2 {3 x - \frac{x^{2}}{2} - \frac{x^{3}}{6} - \frac{3}{2} x}_{0}^{1}$

$\Rightarrow A = 4 - 2 {3 - \frac{1}{2} - \frac{1}{6} - \frac{3}{2}} = \frac{7}{3}$

So, 6A = 14.

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