Maximal $n$-gon in a region
Let $G(n)$ denote the largest possible area of an $n$-gona polygon with $n$ sides contained in the region $\{(x, y) \in \Bbb R^2: x^4 \leq y \leq 1\}$.
For example, $G(3) = 1$ and $G(5)\approx 1.477309771$.
Find $G(101)$ rounded to nine digits after the decimal point.