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From the text beginning with "Let us now figure out the distribution of particles as a function of speed.",

whats the reasoning behind multiplying a radial distribution with a volume?

Is it the same thing as integrating the PDF radially, but only for one instantaneous value instead of carrying out the full integration?

Since we have a radial distribution and a formula relating spherical volume to the radius, shouldn't it be possible to use the PDF or CDF method to derive the volume distribution or is that not what we want?