A395393 Decimal expansion of the volume of the solid of revolution formed by rotating a unit half-width lemniscate of Bernoulli about the horizontal axis.

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

Offset

0,1

Comments

The lemniscate of Bernoulli can be described by (x^2 + y^2)^2 = a^2*(x^2 - y^2) in Cartesian coordinates, or r^2 = a^2*cos(2*phi) in polar coordinates, where a is the half-width. Here we take a = 1.

References

N. P. Bali and Usha Paul, A Textbook of Engineering Mathematics: for B.E., B.Tech. Semester I, Laxmi Publications (P) Ltd, 8th ed., 2014, pp. 142-145.

Links

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

Mandeep Beniwal, Volume of solid of revolution of lemniscate, Smart study helper, YouTube video, 2021.

Amiram Eldar, Illustration of the solid of revolution.

Filo, Find the volume of the solid generated by the revolution of the lemniscate....

Eric Weisstein's World of Mathematics, Lemniscate.

Eric Weisstein's World of Mathematics, Solid of Revolution.

Wikipedia, Lemniscate of Bernoulli.

Wikipedia, Solid of revolution.

Formula

Equals (3*sqrt(2)*arcsinh(1) - 2) * Pi/12.

Equals (3*sqrt(2)*log(1+sqrt(2)) - 2) * Pi/12.

Example

0.455361142379842578562411139680602786173346170098182...

Mathematica

RealDigits[(3*Sqrt[2]*ArcSinh[1] - 2) * Pi/12, 10, 120][[1]]

Prog

(PARI) (3*sqrt(2)*asinh(1) - 2) * Pi/12

Crossrefs

Cf. A091648, A343392, A395392, A395394, A395395.

Sequence in context: A029685 A386738 A153131 * A016718 A165361 A197136

Adjacent sequences: A395390 A395391 A395392 * A395394 A395395 A395396

Keyword

nonn,cons,changed

Author

Amiram Eldar, Apr 21 2026

Status

approved