A396214 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).

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, 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, 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

Offset

0,1

Comments

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).

The volume under the dome is Pi/48.

Links

Table of n, a(n) for n=0..104.

Zarema Seidametova and Valerii Temnenko, Some Geometric Objects Related to a Classical Problem of Galileo, The College Mathematics Journal, Vol. 51, No. 1 (2020), pp. 57-65.

Eric Weisstein's World of Mathematics, Quadrifolium.

Formula

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))).

Example

0.897291807257505298135334708781087967205346224802818...

Mathematica

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

Prog

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

Crossrefs

Cf. A019675, A138500.

Sequence in context: A154842 A394587 A195057 * A395904 A021531 A019695

Adjacent sequences: A396211 A396212 A396213 * A396215 A396216 A396217

Keyword

nonn,cons

Author

Amiram Eldar, May 19 2026

Status

approved