Formula for areas of revolution

Question

The formulas for the area of revolution are:

Now, to find the area of revolution around the x-axis we can use the formula $$S=\int2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx$$ where $y$ is some function of $x$. However, I've been told that you can also $$S=\int2\pi y\sqrt{1+(\frac{dx}{dy})^2}dy$$ to find the same area of revolution around the x-axis, where we're treating $y$ as a variable here. I've been trying to figure out why two integrals are equivalent using substitution but I can't get it to work so I'm just wondering how they're equal and how they both represent the same area of revolution around the x-axis.

Answer 1

you can replace $dy$ in the second integral with $\frac{dy}{dx}dx$ and get the original integral

Answer 2

Note that for the second (rotation about y-axis) we have

$$S=\int2\pi \color{red}x\sqrt{1+(\frac{dx}{dy})^2}dy$$

Note that, in some sense, they are equivalent since one is obtained from the other simply exchanging $x$ with $y$. Notably since the first represent the area of revolution by a rotation about x-axis then the second represent the area of revolution by a rotation about y-axis.