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


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


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


which states the first Pappus’s centroid theorem.

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


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


which may be written shortly

R=12Aab(y22-y12)𝑑x. (2)

The volume of the solid of revolution is


By (2), this attains the form

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