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 _____.

Answer 1

Correct Answer: 5

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:

\[ = \frac{7}{6} - \frac{\pi}{4} \implies n = 5 \]