A395392 - OEIS

A395392

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

%I #11 Jun 28 2026 15:50:57

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

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

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

%N Decimal expansion of the surface area of the solid of revolution formed by rotating a unit half-width lemniscate of Bernoulli about the horizontal axis.

%C 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.

%D J. Dennis Lawrence, A Catalog of Special Plane Curves, Dover Publications Inc., New York, 1972, p. 122.

%D Babu Ram, Engineering Mathematics, Pearson, 2009. See example 8.26.

%H G. N. Berman, <a href="https://archive.org/details/g.-n.-berman-a-problem-book-in-mathematical-analysis-mir-1980/page/177/mode/1up">A Problem Book in Mathematical Analysis</a>, 2nd ed., Mir Publishers, Moscow, 1980, p. 177, Problem 2607 (solution on page 397).

%H Mandeep Beniwal, <a href="https://www.youtube.com/watch?v=lLtiv3iSjVE">Surface of solid of revolution of Lemniscate</a>, Smart study helper, YouTube video, 2021.

%H Amiram Eldar, <a href="/A395392/a395392.png">Illustration of the solid of revolution</a>.

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli">Lemniscate of Bernoulli</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Solid_of_revolution">Solid of revolution</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals 2*Pi*(2 - sqrt(2)).

%F Equals 4*Pi*(1 - 1/sqrt(2)).

%e 3.680604738042440459818811552996624139559434218749042...

%t RealDigits[2*Pi*(2-Sqrt[2]), 10, 120][[1]]

%o (PARI) 2*Pi*(2-sqrt(2))

%Y Cf. A064853, A154739, A154747, A154743, A343392, A395393, A395394, A395395.

%K nonn,cons,changed

%O 1,1

%A _Amiram Eldar_, Apr 21 2026