Touching circle area

Calculus Level 5

Given the function, \[ f(x) = x^n \quad , \quad n \in \mathbb{R}, n \geq 2 \] a circle can be constructed that exactly touches, but not intersects \(f(x)\). See the figure for an example.

\(S\) is the area enclosed by the two curves (gray area in figure). We have \(\displaystyle \lim_{n\rightarrow\infty} S = 1 - \pi / 4\). What is \(S\) for \(n = 3\)? Give your answer to 4 decimal places.

P.S. There is a (not so nice) analytical expression for the area. Can you find it?

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