derivation of Pappus’s centroid theorem


I.  Let s denote the arc rotating about the x-axis (and its length) and R be the y-coordinate of the centroid of the arc.  If the arc may be given by the equation

y=y(x)

where  axb, the area of the formed surface of revolutionMathworldPlanetmath is

A= 2πaby(x)1+[y(x)]2𝑑x.

This can be concisely written

A= 2πsy𝑑s (1)

since differential-geometrically, the product 1+[y(x)]2dx is the arc-element.  We rewrite (1) as

A=s2π1ssy𝑑s.

Here, the last factor is the ordinate of the centroid of the rotating arc, whence we have the result

A=s2πR

which states the first Pappus’s centroid theorem.

II.  For deriving the second Pappus’s centroid theorem, we suppose that the region defined by

axb,0y1(x)yy2(x),

having the area A and the centroid with the ordinate R, rotates about the x-axis and forms the solid of revolutionMathworldPlanetmath with the volume V.  The centroid of the area-element between the arcs  y=y1(x)  and  y=y2(x)  is [y2(x)+y1(x)]/2 when the abscissa is x; the area of this element with the width dx is [y2(x)-y1(x)]dx.  Thus we get the equation

R=1Aaby2(x)+y1(x)2[y2(x)-y1(x)]𝑑x

which may be written shortly

R=12Aab(y22-y12)𝑑x. (2)

The volume of the solid of revolution is

V=πab(y22-y12)𝑑x=A2π12Aab(y22-y12)𝑑x.

By (2), this attains the form

V=A2πR.
Title derivation of Pappus’s centroid theorem
Canonical name DerivationOfPappussCentroidTheorem
Date of creation 2013-03-22 19:36:11
Last modified on 2013-03-22 19:36:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 53A05