>>596

V = ∫[0,1] S(y) dy
 = 2∫[0,1] { (1-yy)arctan(√y) - (1-y)√y } dy
 = 2 [ (1/3)(2-y)(1+y)^2 arctan(√y) - (1/45)(30+35y-21yy)√y ](0,1)
 = 2 (π/3 - 44/45)
 = 0.13883954683764