In this blog, I derived the expression for the surface area of a sphere, . In today’s blog, I will derive the expression for the volume of a sphere. Actually, once one has understood how to derive the surface area of a sphere using spherical polar coordinates, deriving the volume is pretty straight forward. It only involves one extra step, and that is to create a volume element with the same surface area that we had before, but with a thickness , and to integrate over in addition to integrating over .
As we can see from the figure below, the volume element is given by where is the same surface element we derived before, namely . So, the expression for our volume element is
We need to integrate this volume element over all three variables so we have