OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Bean Curve
EXAMPLE
3.750213645...
MATHEMATICA
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]
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 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jul 29 2011
STATUS
approved