From: Clive Tooth 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/ End of document