If the area of the region $$S = \{(x,y): 2y - y^2 \le x^2 \le 2y, x \ge y\}$$ is equal to $$\dfrac{n+2}{n+1} - \dfrac{\pi}{n-1}$$, then the natural number $$n$$ is equal to ______.

Calculation:

$$x^2+y^2-2y\ge0 \ \&\ x^2-2y\le0,\ x\ge y$$

Hence required area

$$=\frac12\times2\times2-\int_0^2\frac{x^2}{2}\,dx-\left(\frac{\pi}{4}-\frac12\right)$$

$$=\frac76-\frac{\pi}{4}$$

Comparing with

$$\frac{n+2}{n+1}-\frac{\pi}{n-1}$$

we get

$$n=5$$

Hence, the correct answer is $$5$$.