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

Regarding Don P's calculation:

I integrated the volume of a parabola rotated about the y-axis with
R_1 the bottom (larger)radius of revolution, R_2 the top (smaller)
radius of revolution, and h = vertical distance between the bottom and top:

Parabola: x = a*y^2 + c
a = (R_2 - R_1)/h^2
c = R_1
V= pi*h*((3*R_2^2 + 4*R_1*R_2 + 8*R_1^2)/15)
V_cyl = pi*h*R^2
R = sqr((3*R_2^2 + 4*R_1*R_2 + 8*R_1^2)/15)

Using Don's definitions for his truncated paraboloid of revolution I get:

V= π*h*((R_2^2 + 8*R_1*R_2 + 6*R_1^2)/15)

And thus

R_avg = √[(R_2^2 + 8*R_1*R_2 + 6*R_1^2)/15].

BTW, my calculations do match Don's for his sphere truncated at 2 different latitudes. But that situation has the figure containing a central convex equatorial bulge larger than either of the truncated radii, each of which are in *different* hemispheres, and that makes the so-called average *not* have the important central tendency property that any respectable or self-respecting average ought to possess, i.e. Min(R_1, R_2) ≤ R_avg ≤ Max(R_1, R_2).

David Bowman