The Area of the Surface Generated by Revolving a Graph About Any Line

TL;DR

This paper derives a general formula for calculating the surface area generated by revolving a graph around any line, extending classical surface area formulas to more general axes of revolution.

Contribution

It introduces a comprehensive formula for surface area generated by revolution around arbitrary lines, generalizing existing formulas for specific axes.

Findings

Derived a general formula for surface area of revolution around any line.

Obtained a specific formula for revolution around lines of the form y = m x + k.

Extended classical surface area formulas to more general axes.

Abstract

We discuss a general formula for the area of the surface that is generated by a graph $[t_0, t_1] \to \mathbb R^2$ sending $t \mapsto \bigl(x(t), y(t) \bigr)$ revolved around a general line $L: A x + B y = C$. As a corollary, we obtain a formula for the area of the surface formed by revolving $y = f(x)$ around the line $y = m x + k$.