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%I #35 Jul 25 2020 10:21:54
%S 3,7,5,0,2,1,3,6,4,5,1,5,7,2,4,2,5,7,1,9,2,8,2,9,5,7,9,6,6,0,5,5,1,4,
%T 0,3,1,6,1,8,2,4,5,4,8,9,8,5,1,0,4,9,1,3,0,6,0,5,0,7,8,5,9,7,8,3,9,2,
%U 0,3,0,5,9,5,5,9,8,1,4,3,1,3,0,5,7,4,2,4,8,0,2,3,2,7,9,6,2,2,6,5,1,5,9,8,6,1,8,5,7,4
%N Decimal expansion of bean curve perimeter.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeanCurve.html">Bean Curve</a>
%e 3.750213645...
%t f[x_, y_] = x^4 + x^2*y^2 + y^4 - x*(x^2 + y^2); x1 = 1/3; x2 = 5/6; sx = Solve[f[x, y] == 0, x]; sy = Solve[f[x, y] == 0, y]; g1[y_] = x /. sx[[3]]; g2[y_] = x /. sx[[4]]; f[x_] = y /. sy[[4]]; p1 = NIntegrate[ Sqrt[1 + g1'[y]^2], {y, 0, f[x1]}, WorkingPrecision -> 120]; p2 = NIntegrate[ Sqrt[1 + f'[x]^2], {x, x1, x2}, WorkingPrecision -> 120]; p3 = NIntegrate[ Sqrt[1 + g2'[y]^2], {y, 0, f[x2]}, WorkingPrecision -> 120]; Take[ RealDigits[2*(p1+p2+p3)][[1]], 105]
%t Take[RealDigits[9/2 + NIntegrate[2 Sqrt[1 + (1 - 2 x + (1 + 3 x - 6 x^2)/Sqrt[1 + (2 - 3 x) x])^2/(8 x (1 - x + Sqrt[1 + (2 - 3 x) x]))] - 1/Sqrt[x] - 1/(2 (1 - x)^(3/4)) - 3/(8 (1 - x)^(1/4)), {x, 0, 1}, WorkingPrecision -> 220, PrecisionGoal -> 110, MaxRecursion -> 50]][[1]], 110] (* _Eric W. Weisstein_, Jul 23 2020 *)
%Y Cf. A193505 (area).
%Y Cf. A336501 (decimal expansion of the lima bean curve).
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Jul 29 2011