different ways to calculate the volume of a ball with radius $R$

Question

We all know the volume is $\dfrac{4\pi R^3}{3}$. the 2 regular of ways to reach that are :
(1) triple integral (there are at least 2 ways of solving the integral).
(2) Solid of revolution.

I have to guess that there are more ways of calculation..

Answer 1

You can use shell method or washer method for solid revolution.

You can use cartesian, cylindrical, spherical coordinates systems for triple integral.

You can consider A sphere to be made up of layers of spherical shells with area $4\pi r^2$ and then integrate it, which is a more intuitive way of doing the triple integral in spherical coordinates.

You can solve the differential equation to find the largest volume that can be contained by a chosen surface area (would be quite difficult)

Answer 2

You could consider the n-dimensional volume of an n-dimensional hypersphere and then differentiate.