# Superb Circular Shapes

Calculus Level 3

In this problem I asked your attention for the curve $S:\ \sqrt{1-|x|} + \sqrt{1-|y|} = 1,$ which looks a lot like a circle but bulges just a little. It is easy to check that this curve runs through the points $$(0,\pm 1),\ (\pm 1,0)$$, as well as $$\big(\pm\tfrac34,\pm\tfrac34\big)$$.

As someone pointed out, this has to do with superellipses--but curve $$S$$ is not actually a superellipse. However, there exists a superellipse that is very similar to our shape, given by $T:\ |x|^n + |y|^n = 1$ (with $$n$$ a real parameter), which runs through the same eight points $$(0,\pm 1),\ (\pm 1,0)$$, as well as $$\big(\pm\tfrac34,\pm\tfrac34\big)$$.

A section of both curves is shown in the diagram above: the blue graph is curve $$S,$$ and the orange graph the superellipse $$T$$.

Clearly, the nearly circular area included by $$T$$ is slightly larger than that included by $$S$$. How much bigger is it, expressed as a percentage?

Notes:

• This problem requires numerical integration. (If you can find a different solution, I am very interested!)
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