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2) Flared (Parabolic) Horn
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)
For a similar case, David Bowman got:
R = sqr((R_2^2 + R_2^3/2 * R_1^1/2 + R_2* R_1 + R_2^1/2 * R_1^3/2 + R_1^2)/5)
This does not agree with my expression. However, both expressions
yield the correct result in the limiting case when a = 0 (i.e. R_2 = R_1). So,
the "parabola" is simply a straight vertical line rotated about the y=axis
and the volume is that of a right circular cylinder with radius R = R_1 = R_2.
I assume this means that David's parabola choice is somehow different
from mine and both results are correct.