From: Clive Tooth <clive@pisquaredoversix.force9.co.uk>
Subject: Re: Second Theorem of Pappus, Centroid of a Curve
Date: Sat, 06 Mar 1999 22:56:22 +0000
Newsgroups: sci.math

Chris Subich wrote:

> Howdy, all.
> 
> This afternoon, my math class was attempting to use the Second Theorem
> of Pappus (Surfaces of Revolution), when we hit a dead-end--Our
> interpretation didn't work.
> 
> In the few books that we have that even devote a paragraph to the
> topic (2/6, I believe), they mention something about the centroid of a
> curve.
> 
> Which brings me to my question...  What exactly is that, and how is it
> related to the centroid of a (2D) region?
> 
> Also, if anyone could give me a brief working formula, etc., I can be
> sure that the interpretation isn't off, either.
> 
> Please send a copy of your reply to csubich@ibm.net, so I won't miss
> it.

The centroid of a curve is just the center of gravity of a piece of wire
bent into the shape of the curve. It has no particular relationship,
afaik, with the centroid of a 2D region. The two theorems of Pappus, in
this part of mathematics, are:

1) The surface area of a solid of revolution is equal to the length of
the curve being rotated TIMES the distance traveled by the centroid of
the 1-D curve. This assumes that the curve being rotated does not cross
the axis of rotation.

2) The volume of a solid of revolution is equal to the area of the shape
being rotated TIMES the distance traveled by the centroid of the 2-D
shape. This assumes that the shape being rotated does not cross the axis
of rotation.

Formula for the y-coordinate of the centroid of a 1-D curve:

Integral from a to b y ds   Total moment of the curve
------------------------- = -------------------------
Integral from a to b ds     Length of the curve

where ds is the arc-length differential.

Formula for the y-coordinate of the centroid of a 2-D shape:

Integral from a to b y^2/2 dx   Total moment of the shape
----------------------------- = -------------------------
Integral from a to b y dx       Area of the shape

Here is a trivial example: Let y=mx, we will generate a cone of height
h.

Length of arc = sqrt(h^2+h^2m^2)
Distance of (1-D) centroid from x-axis = hm/2
Surface area = sqrt(h^2+h^2m^2) * 2pi * hm/2 = pi*r*s [well known
formula where r=radius of base(=hm) and s is the slant height]

Area of shape being rotated = h^2m/2
Distance of (2-D) centroid from x-axis = hm/3
Volume = h^2m/2 * 2pi * hm/3 = pi*r^2*h/3 [well known formula]

Please post again if you need more help.

-- 
 
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
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