JEE Mains · Maths · STD 12 - 8. Application and integration
If the area of the region \(S=\left\{(x, y): 2 y-y^2 \leq x^2 \leq 2 y, x \geq y\right\}\) is equal to \(\frac{ n +2}{ n +1}-\frac{\pi}{ n -1}\), then the natural number \(n\) is equal to \(...............\).
- A \(4\)
- B \(3\)
- C \(2\)
- D \(5\)
Answer & Solution
Correct Answer
(D) \(5\)
Step-by-step Solution
Detailed explanation
\(x^2+y^2-2 y \geq 0 \ x^2-2 y \leq 0, x \geq y\) Hence required area \(=\frac{1}{2} \times 2 \times 2-\int_0^2 \frac{x^2}{2} d x-\left(\frac{\pi}{4}-\frac{1}{2}\right)\) \(=\frac{7}{6}-\frac{\pi}{4} \Rightarrow n=5\)
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