Approximating volume of a sphere using spherical cones in the limit

Question

[Disclaimer: I am not a mathematician, please bear my naive wordings]

I am looking for a reference that can give me a proof to my simple question: Can we approximate the volume of a sphere using spherical cones emanating from the center of the sphere, in the limit (-> infinity) ?

In the limit, the cone size goes infinitely small and each cone can be considered as a Dirac-beam.

I have a feeling that this is true but I am looking for a good reference that could explain most of the steps in the proof.

Answer 1

The infinitesimal cone is not really a Dirac-beam since its volume is a third of base area $A$ times altitude $h$, namely $\frac{1}{3}Ah$, rather than the product $Ah$. The argument is valid and can be formalized in the context of true infinitesimal calculus; see Elementary Calculus.