A395393 - OEIS

%I #8 Jun 28 2026 15:50:57

%S 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,

%T 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,

%U 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

%N Decimal expansion of the volume 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 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.

%H Mandeep Beniwal, <a href="https://www.youtube.com/watch?v=MBXioI6FQhM">Volume of solid of revolution of lemniscate</a>, Smart study helper, YouTube video, 2021.

%H Amiram Eldar, <a href="/A395393/a395393.png">Illustration of the solid of revolution</a>.

%H Filo, <a href="https://askfilo.com/user-question-answers-smart-solutions/find-the-volume-of-the-solid-generated-by-the-revolution-of-3332363435303032">Find the volume of the solid generated by the revolution of the lemniscate...</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>.

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

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

%e 0.455361142379842578562411139680602786173346170098182...

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

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

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

%K nonn,cons,changed

%O 0,1

%A _Amiram Eldar_, Apr 21 2026