Is double integration an easier way to find volume of rotation?

Question

AP Calculus BC student here,

One of the most hated topics from Calculus 1 & 2 is often the disk method, washer method, and the shell method.

Disk Method = $\pi \int [f(x)^2]dx$ (rotate x-axis)

Washer Method = $\pi \int [R(x)^2-r(x)^2]dx$ (rotate x-axis)

Shell Method = $2\pi \int xf(x)dx$ (rotate y-axis)

Is there a method from multivariable calculus that uses double integration to calculate the volume of rotation?

Answer 1

If I understand correctly, you are complaining that there are three methods to bear in mind when determining the volume of a body of revolution. And that, in itself, does not cover the full gamut of possibilities. I feel your pain; years ago I adopted the Pappus centroid theorems for volume and surface area of bodies of revolution.

Pappus's ($2^{nd}$) Centroid Theorem: the volume of a planar area of revolution is the product of the area $A$ and the length of the path traced by its centroid $R$, i.e., $2\pi R$. The bottom line is that the volume is given simply by $V=2\pi RA$.

And while we're here, Pappus's ($1^{st}$) Centroid Theorem: the surface area of a planar area of revolution is the product of the curve length $L$ and the length of the path traced by its centroid $R$, thus, $S=2\pi RL$.

There is no ambiguity in determining the area and centroid. (Note that the centroid of the area and line curve are not the same.)

Answer 2

These really are double integrals already. They look like single integrals because one of the integrals is the integral that calculates the area or circumference of a circle, for which you have formulas. The factor of $\pi$ is a clue to that.