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Decimal expansion of the area of the ampersand curve.
1

%I #16 Aug 15 2015 13:12:01

%S 1,0,6,6,5,5,5,0,9,5,4,5,7,3,2,7,4,6,3,7,5,3,9,1,3,9,1,3,3,6,9,1,1,9,

%T 1,3,5,7,5,5,3,9,2,3,3,9,8,4,7,7,4,9,8,9,0,2,3,7,1,7,7,0,4,4,6,4,0,4,

%U 9,8,9,3,5,1,5,1,7,9,3,2,4,8,3,8,1,1,8,0,2,1,8,7,7,1,9,9,7,2,5,7,3,0,6,6,3

%N Decimal expansion of the area of the ampersand curve.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AmpersandCurve.html">Ampersand Curve</a>

%e 1.06655509...

%t eq = (y^2 - x^2)(x - 1)(2x - 3) == 4 (x^2 + y^2 - 2x)^2 ;

%t sy = Solve[eq, y]; f1[x_] = y /. sy[[2]]; f2[x_] = y /. sy[[4]]; x1 = x /. FindRoot[f1'[x] == 1, {x, 31/21}, WorkingPrecision -> 120 ]; y1 = y /. Solve[eq /. x -> x1][[3]]; y2 = y /. Solve[eq /. x -> x1][[4]]; sx = Solve[eq, x]; g1[y_] = x /. sx[[1]]; g2[y_] = Simplify[x /. sx[[4]], y1 < y < y2]; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Simplify[-g1[y], 0 < y < Sqrt[3]/2] , {y, 0, Sqrt[3]/2}]; i2 = ni[f2[x] - f1[x], {x, 0, 1}]; i3 = ni[f2[x] - f1[x], {x, 1, x1}]; i4 = ni[g2[y] - x1 , {y, y1, y2}]; Take[RealDigits[2 (i1 + i2 + i3 + i4)][[1]], 105]

%t (* _Jean-François Alcover_, Aug 03 2011 *)

%K nonn,cons

%O 1,3

%A _Eric W. Weisstein_, Dec 16 2004