Question:
If the area of the region \( S={(x,y) : 2y-y^2 ≤ x^2 ≤ 2y, x ≥ y}\) is equal to \(\frac{ n+2}{n+1}−\frac{π}{n−1}\), then the natural number n is equal to _____.
If the area of the region \( S={(x,y) : 2y-y^2 ≤ x^2 ≤ 2y, x ≥ y}\) is equal to \(\frac{ n+2}{n+1}−\frac{π}{n−1}\), then the natural number n is equal to _____.
Updated On: Feb 20, 2026
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Correct Answer: 5
Solution and Explanation
strong>Given Function:
\(x^2 + y^2 - 2y \geq 0 \) & \(x^2 - 2y \leq 0\), \(x \geq y\)
Required Area Calculation:
\[ \text{Required Area} = \frac{1}{2} \times 2 \times 2 - \int_{2}^{2} \frac{x^2}{2} dx - \frac{\pi}{4} - \frac{1}{2} \]
Simplifying:
\[ = \frac{7}{6} - \frac{\pi}{4} \implies n = 5 \]
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