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 a≤x≤b, the area of the formed surface of revolution is
A= 2π∫bay(x)√1+[y′(x)]2𝑑x. |
This can be concisely written
A= 2π∫sy𝑑s | (1) |
since differential-geometrically, the product √1+[y′(x)]2dx is the arc-element. We rewrite (1) as
A=s⋅2π⋅1s∫sy𝑑s. |
Here, the last factor is the ordinate of the centroid of the rotating arc, whence we have the result
A=s⋅2πR |
which states the first Pappus’s centroid theorem.
II. For deriving the second Pappus’s centroid theorem, we suppose that the region defined by
a≤x≤b,0≤y1(x)≤y≤y2(x), |
having the area A and the centroid with the ordinate R, rotates about the x-axis and forms the solid of revolution 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=1A∫bay2(x)+y1(x)2[y2(x)-y1(x)]𝑑x |
which may be written shortly
R=12A∫ba(y22-y21)𝑑x. | (2) |
The volume of the solid of revolution is
V=π∫ba(y22-y21)𝑑x=A⋅2π⋅12A∫ba(y22-y21)𝑑x. |
By (2), this attains the form
V=A⋅2πR. |
Title | derivation of Pappus’s centroid theorem |
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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 |