I'm trying to wrap my head around the two-dimension recursion formula for which there are few threads here, but which weren't clarifying.
The Wikipedia article speaks of:
The volume of the ball can therefore be written as an iterated integral of the volumes of the (n − 2)-balls over the possible radii and azimuths.
Why are the $(n-2)$ balls iterated over radii and azimuths?
I tried to experiment with
$$\frac{2\pi r^2}{n}V_{n-2}(R)$$
which for $n=4$:
$$\frac{2\pi r^2}{4}\pi r^2=\frac{\pi^2r^4}{2}$$
which is the volume of a 4-sphere.
But I don't understand the integrations and why the go from $0$ to $2 \pi$ and $0$ to $r$. Nor why the integral is claimed to be the same for all $V_n$s and $V_{n-2}$s (surely the elements that are integrated are of different shape?).
I.e. why is the volume integral
$$V_4 (R) = \int_0^{2\pi} \int_0^R V_{n-2}$$
Also what is the $r$ in
$$\int_0^{2\pi} \int_0^R V_{n-2}(\sqrt{R^2-r^2})\,\color{red}r\, dr d\theta$$