Let the area enclosed between the curves and be . If are integers, then the value of equals [2025]

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Solution

Q.
Let the area enclosed between the curves $| y | = 1 - x^{2}$ and $x^{2} + y^{2} = 1$ be $α$ . If $9 α = β π + γ; β, γ$ are integers, then the value of $| β - γ |$ equals [2025]

1 27

2 15

3 33

4 18

Ans.
(3)

We have, $c_{1} : | y | = 1 - x^{2}$ and $c_{2} : x^{2} + y^{2} = 1$

$∴$ Required area $= α = 4 [(\begin{matrix} Area of circle in \\ 1^{st} quadrant \end{matrix}) - \int_{0}^{1} (1 - x^{2}) d x]$

$= 4 [(\frac{π (1)}{4}) - {[x - \frac{x^{3}}{3}]}_{0}^{1}]$

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

$= π - \frac{8}{3}$

Hence, $9 α = 9 π - 24$

On comparing with $9 α = β π + γ$ , we get $β = 9$ and $γ = - 24$

$∴ | β - γ | = | 9 - (- 24) | = 33$ .

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