A396214 - OEIS

%I #5 May 19 2026 06:55:33

%S 8,9,7,2,9,1,8,0,7,2,5,7,5,0,5,2,9,8,1,3,5,3,3,4,7,0,8,7,8,1,0,8,7,9,

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

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

%N Decimal expansion of the surface area of the roof of Galileo's dome, a surface that can be described in cylindrical coordinates by z = r * tan(phi) - r^2/(2*cos(phi)^2).

%C This constant is the area of the roof, while the base of the dome is the interior of the folium of Galileo, one of the four leaves of the quadrifolium, whose area is Pi/8 (A019675).

%C The volume under the dome is Pi/48.

%H Zarema Seidametova and Valerii Temnenko, <a href="https://www.jstor.org/stable/48662087">Some Geometric Objects Related to a Classical Problem of Galileo</a>, The College Mathematics Journal, Vol. 51, No. 1 (2020), pp. 57-65.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Quadrifolium.html">Quadrifolium</a>.

%F Equals Integral_{x=0..Pi/2} f(cos(x)^2) dx, where f(x) = sqrt(1 - 3*x + 4*x^2) * (5 - 9*x + 5*x^2 + 3*x^3) / 6 - (1 + x) * (x^2 - 1/3) / 2 + (1/2) * x^2 * (1 + x) * sqrt(1 - x) * log(((1 - x)^(3/2) + sqrt(1 - 3*x + 4*x^2)) / (sqrt(1 + x) - (1 + x) * sqrt(1 - x))).

%e 0.897291807257505298135334708781087967205346224802818...

%t f[x_] := Sqrt[1 - 3*x + 4*x^2] * (5 - 9*x + 5*x^2 + 3*x^3) / 6 - (1 + x) * (x^2 - 1/3) / 2 + (1/2) * x^2 * (1 + x) * Sqrt[1 - x] * Log[((1 - x)^(3/2) + Sqrt[1 - 3*x + 4*x^2]) / (Sqrt[1 + x] - (1 + x) * Sqrt[1 - x])];

%t $MaxExtraPrecision = 1000; RealDigits[NIntegrate[f[Cos[x]^2], {x, 0, Pi/2}, WorkingPrecision -> 120]][[1]]

%o (PARI) f(x) = sqrt(1 - 3*x + 4*x^2) * (5 - 9*x + 5*x^2 + 3*x^3) / 6 - (1 + x) * (x^2 - 1/3) / 2 + (1/2) * x^2 * (1 + x) * sqrt(1 - x) * log(((1 - x)^(3/2) + sqrt(1 - 3*x + 4*x^2)) / (sqrt(1 + x) - (1 + x) * sqrt(1 - x)));

%o intnum(x = 0, Pi/2, f(cos(x)^2))

%Y Cf. A019675, A138500.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, May 19 2026