Integer Integral

The area function in question computes to:

$f(n) = \int_{-\sqrt{n}}^{\sqrt{n}} n - x^2 dx = nx - \frac{x^3}{3}|_{-\sqrt{n}}^{\sqrt{n}} = \frac{4}{3} \cdot n^{\frac{3}{2}}$

and its inverse is:

$f^{-1}(n) = (\frac{3}{4} \cdot n )^{\frac{2}{3}}$ .

For $a \in \mathbb{N}$ , the smallest input where $f(a) \in \mathbb{N}$ occurs at $a = 9 \Rightarrow f(9) = \frac{4}{3} \cdot 9^{\frac{3}{2}} = 36$ . Thus, $f^{-1}(9) = (\frac{3}{4} \cdot 9)^{\frac{2}{3}} = \frac{9}{16^{\frac{1}{3}}} \approx \boxed{3.57}.$