Superb Circular Shapes

Calculus Level pending

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:

  • Give your answer as a percentage with three decimals precision.
  • This problem requires numerical integration. (If you can find a different solution, I am very interested!)
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