# 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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