Formula of Volume of Revolution with Integration by Parts and Extension
This provides a mathematical tool for computing volumes in calculus, but it is incremental as it builds on existing integration techniques.
The paper derived a formula for calculating volumes of revolution using integration by parts for monotone functions and extended it to continuous, piecewise strictly monotone, and differentiable functions, with examples applied to Kepler equation-based curvilinear trapezoids.
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.