Formula of Volume of Revolution with Integration by Parts and Extension

TL;DR

This paper derives a formula for calculating the volume of revolution using integration by parts, extending it to cases involving curved trapezoids defined by monotone functions, with practical examples involving Kepler's equation.

Contribution

It introduces a new calculation formula for volumes of revolution that extends traditional methods to more complex curved trapezoids defined by monotone functions.

Findings

Derived a volume formula using integration by parts.

Extended the formula to curved trapezoids from monotone functions.

Provided examples involving Kepler's equation.

Abstract

A calculation formula of volume of revolution with integration by parts of definite integral is derived based on monotone function, and extended to a general case that curved trapezoids is determined by continuous, piecewise strictly monotone and differential function. And, two examples are given, ones curvilinear trapezoids is determined by Kepler equation, and the other curvilinear trapezoids is a function transmuted from Kepler equation.