If the area of the region is , then is equal to _________. [2025]

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Solution

Q.
If the area of the region ${(x, y) : | 4 - x^{2} | \leq y \leq x^{2}, y \leq 4, x \geq 0}$ is $(\frac{80 \sqrt{2}}{α} - β), α, β \in N$ , then $α + β$ is equal to _________. [2025]

Ans.
(22)

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

$= {[\frac{2 x^{3}}{3} - 4 x]}_{\sqrt{2}}^{2} + 8 \sqrt{2} - 8 - {[\frac{x^{3}}{3} - 4 x]}_{2}^{2 \sqrt{2}}$

$= \frac{40 \sqrt{2}}{3} - 16 = \frac{80 \sqrt{2}}{6} - 6$

$\Rightarrow α = 6, β = 16$

$\Rightarrow α + β = 22$ .

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