For the curious, in order to understand what 4 dimensional volume has to do with physics think about the idea of work, as the integral of the product of force acting in a given direction times mass. If density is assumed constant this amounts to multiplying distance times volume, a one dimensional concept times a three dimensional one, or a 4 dimensional quantity. This is why measuring work is essentially the same as measuring 4 dimensional volume, and this is the explanation of why Archimedes' arguments, which were based on physics, yield a nice computation of the volume of the 4 dimensional ball.
