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[Phys-L] rms / conic / arithmetic / geometric averages

Hi --

I recently noticed something cute about averages. Maybe everybody
but me already knew this, but I was surprised.

Context: Suppose you have a round pot or bucket with uniformly
tapering sides. In other words, it is a truncated cone. You know
the height, and you know the ID at the bottom and at the top.

We all know the formula for the volume of a cylinder, and this
is "almost" a cylinder, so maybe we can use the same formula,
using some sort of *average* radius.

What sort of average should we use? RMS? Arithmetic? Geometric????

For a full cone, we know the volume is 1/3 of the volume of a
cylinder with the same diameter and radius. So RMS is no good,
because that would predict 1/2 of the volume rather than 1/3.
And arithmetic is no good, because it would predict 1/4 of the
volume (half of the diameter) instead of 1/3 of the volume.
Geometric is even worse, because it would predict 0 instead
of 1/3. We want something that is somewhere between RMS and

The cute thing is that all of the above form a family. You can
interpolate between them. The general formula is:
average = √[(Bot2+N*Bot*Top+Top2) / (2+N)]

for all N from 0 to ∞. In particular:

N=0 --> RMS
N=1 --> conic
N=2 --> arithmetic
N=∞ --> geometric

The relationships are graphed here:

The N=1 average is in fact exactly correct for a truncated cone,
as you can easily verify in lots of ways. (Grind out the integrals.
Or just subtract one full cone from another.)

Related: Here is an online calculator to figure out the volume
of a truncated cone: