Tilted Cone and Cylinder, Cone and Tilted Sphere

TL;DR

This paper revisits classical volume problems involving elliptic integrals, providing detailed derivations and approximations for complex formulas related to tilted cones, cylinders, and spheres.

Contribution

It offers detailed derivations of known formulas and introduces useful approximations for complex volume calculations involving elliptic integrals.

Findings

Neat formula for the volume of a tilted cone using elliptic integrals

Messy but useful formula for a related volume problem

Effective approximation methods for complex volume formulas

Abstract

In this note, we will consider two classical volume problems related to elliptic integrals. The first problem has a neat formula by means of elliptic integrals. We remade it with details. In the second problem, we found a messy formula. On the other hand, it seems to be useful to find a good approximation for the volume.