<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4<italic>/</italic>3 of the area of a certain inscribed triangle. In this paper we generalize this theorem to the <italic>n</italic>-dimensional euclidean space, <italic>n</italic> ≥ 3. It appears that the ratio of the volume of an <italic>n</italic>-dimensional solid bounded by an (<italic>n</italic> − 1)-dimensional hyper-paraboloid and an (<italic>n</italic> − 1)-dimensional hyperplane and the volume of a certain inscribed cone (we analogously repeat Archimedes’ procedure) depends only on the dimension of the euclidean space and it equals to 2<italic>n/</italic>(<italic>n</italic> +1).</p>
</abstract>

Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4/3 of the area of a certain inscribed triangle. In this paper we generalize this theorem to the n-dimensional euclidean space, n ≥ 3. It appears that the ratio of the volume of an n-dimensional solid bounded by an (n − 1)-dimensional hyper-paraboloid and an (n − 1)-dimensional hyperplane and the volume of a certain inscribed cone (we analogously repeat Archimedes’ procedure) depends only on the dimension of the euclidean space and it equals to 2n/(n +1).

A Generalization of Archimedes’ Theorem on the Area of a Parabolic Segment

The John Paul II Catholic University of Lublin ul

Department of Applied Mathematics and Computer Science

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"A Generalization of Archimedes\u2019 Theorem on the Area of a Parabolic Segment"}

Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4/3 of the area of a certain inscribed triangle. In this paper we...

A Generalization of Archimedes’ Theorem on the Area of a Parabolic Segment

Pubblicato online: 08 lug 2021

Pagine: 199 - 209

Ricevuto: 24 set 2020

Accettato: 15 nov 2020

DOI: https://doi.org/10.2478/auom-2021-0026

Parole chiave
Archimedes’ theorem, parabolic segment, multiple integral, spherical coordinates

© 2021 Armen Grigoryan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

A Generalization of Archimedes’ Theorem on the Area of a Parabolic Segment

Pubblicato online: 08 lug 2021

Pagine: 199 - 209

Ricevuto: 24 set 2020

Accettato: 15 nov 2020

DOI: https://doi.org/10.2478/auom-2021-0026

Parole chiaveArchimedes’ theorem, parabolic segment, multiple integral, spherical coordinates

© 2021 Armen Grigoryan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Parole chiave
Archimedes’ theorem, parabolic segment, multiple integral, spherical coordinates