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 Square the Circle (Posted on 2004-10-28)
There is a family of curves on the Cartesian plane described by this form:

If n is equal to 2, then it describes an ellipse (if a = b, then it describes a circle).

If n is greater than 2, then this is a "superellipse" (if a = b, then this is a supercircle).

As n increases, the ellipse becomes more "rectangularish", and as n approaches ∞, the limit is a rectangle (or a square if a=b).

What value must n have such that the figure has an area exactly halfway between the associated ellipse (when n=2) and rectangle (when n=∞)?

The graphs below, calculated by varying n with a = b = 1, show this property. Note that as n approaches zero, the curve degenerates into two crossed lines along the x- and y-axes.

 No Solution Yet Submitted by SilverKnight Rating: 3.0000 (4 votes)

Comments: ( You must be logged in to post comments.)
 Subject Author Date sol using mathematica sunny 2004-10-31 11:57:12 Riemann sums Charlie 2004-10-29 02:11:20 re: No Subject (numerically solved) Christian 2004-10-28 21:00:30 No Subject Christian 2004-10-28 20:37:23 re: Some thoughts--Numeric solution Charlie 2004-10-28 19:44:43 Some thoughts Federico Kereki 2004-10-28 18:56:10 re: Quad I Charlie 2004-10-28 18:24:25 Quad I nikki 2004-10-28 18:05:13
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