A122952 -id:A122952 - cp's OEIS frontend

A265729 Decimal expansion of 32*Pi.

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

Offset: 3

Eric W. Weisstein and Robert G. Wilson v, Dec 14 2015

"The integral corresponds to integration over a spherical cone with opening angle Pi/2 and radius 4. However, it is not clear what the integrand physically represents (it resembles computation of a moment of inertia, but that would give a factor (rho*sin(phi))^2 rather than the given rho*cos(phi))."

			100.53096491487338363080458826494409229430942078000338627119822695385012500...

The Jun 02 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007)

Eric Weisstein's World of Mathematics, Definite Integral
Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721, A228824, A229939, A061146.

RealDigits[32 Pi, 10, 111][[1]] (* or *)
Integrate[\[Rho] Cos[\[Phi]] \[Rho]^2 Sin[\[Phi]], {\[Rho], 0, 4}, {\[Phi], 0, Pi/4}, {\[Theta], 0, 2 Pi}]

PARI
```
32*Pi \\ Altug Alkan, Dec 14 2015
```

Equals Integral_{theta=0..2*Pi} Integral_{phi=0..Pi/4} Integral_{rho=0..4} (rho*cos(phi))*rho^2*sin(phi) d(rho) d(phi) d(theta).

A359104 Decimal expansion of the area enclosed by Sylvester's Bicorn curve.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Dec 18 2022

The Cartesian equation of Sylvester's Bicorn curve is y^2*(m^2-x^2) = (x^2+2*m*y-m^2)^2, here with parameter m=1. The area is proportional to the square m^2 of parameter m.

Corresponding arc length is given by A228764.

			0.746455945439346463341461672758965758770535375107896820343...

M. Protat, Des Olympiades à l'Agrégation, Encadrement du bicorne, Problème 66, pp. 142-145, Ellipses, Paris 1997.

Robert Ferréol, Bicorn, Mathcurve.
Eric Weisstein's World of Mathematics, Bicorn.
Wikipedia, Bicorn.
Index to sequences related to curves.

Cf. A228764 (length).

Other area of curves: A019692 (deltoid), A197723 (cardioid), A122952 (nephroid), A180434 (Newton strophoid).

Maple
```
evalf((16*sqrt(3) - 27)*Pi/3, 100);
```

RealDigits[(16*Sqrt[3] - 27)*Pi/3, 10, 120][[1]] (* Amiram Eldar, Dec 18 2022 *)

Equals (16*sqrt(3) - 27)*Pi/3.

cp's OEIS Frontend

A265729 Decimal expansion of 32*Pi.

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