Editing Pappus's centroid theorem

{{Short description|Results on the surface areas and volumes of surfaces and solids of revolution}}
[[File:Pappus centroid theorem areas.gif|thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance ''a'' (in red) from the axis of rotation.]]

In mathematics, '''Pappus's centroid theorem''' (also known as  the '''Guldinus theorem''', '''Pappus–Guldinus theorem''' or '''Pappus's theorem''') is either of two related [[theorem]]s dealing with the [[surface area]]s and [[volume]]s of [[surface of revolution|surface]]s and [[solid of revolution|solid]]s of revolution.

The theorems are attributed to [[Pappus of Alexandria]]{{efn|See:<ref>{{cite book| author=Pappus of Alexandria|author-link=Pappus of Alexandria|editor-last=Jones| editor-first=Alexander| year=1986| title=Book 7 of the ''Collection''| volume=8|location=New York | publisher=Springer-Verlag |isbn=978-1-4612-4908-5 |doi=10.1007/978-1-4612-4908-5 |orig-year=c. 320|series=Sources in the History of Mathematics and Physical Sciences}}</ref>
{{Quote|
They who look at these things are hardly exalted, as were the ancients and all who wrote the finer things. When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers:<br>

The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes. These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the ''First Elements''.
 |author=Pappus
 |source=''Collection'', Book VII, ¶41‒42}}
}} and [[Paul Guldin]].{{efn|"Quantitas rotanda in viam rotationis ducta, producit Potestatem Rotundam uno gradu altiorem, Potestate sive Quantitate rotata."<ref>{{cite book|author-last=Guldin|author-first=Paul|author-link=Paul Guldin|title=De centro gravitatis trium specierum quanitatis continuae| volume=2| pages=147| year=1640 |publisher=Gelbhaar, Cosmerovius | location=Vienna | url=https://books.google.com/books?id=CNaI61CYc94C&pg=PA147|access-date=2016-08-04}}</ref>
That is: "A quantity in rotation, multiplied by its circular trajectory, creates a circular power of higher degree, power, or quantity in rotation."<ref name="RdG2015">{{cite book|author-last=Radelet-de Grave|author-first=Patricia |editor-last=Jullien|editor-first=Vincent |title=Seventeenth-Century Indivisibles Revisited|chapter=Kepler, Cavalieri, Guldin. Polemics with the departed |pages=68 |publisher=Birkhäuser |series=Science Networks. Historical Studies |volume=49 |isbn=978-3-3190-0131-9 |chapter-url=https://books.google.com/books?id=8Vt1CQAAQBAJ&pg=PA68 |date=2015-05-19 |doi=10.1007/978-3-319-00131-9 |issn=1421-6329 |location=Basel |hdl=2117/28047 | access-date=2016-08-04}}</ref> }} Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.<ref>{{cite journal |last1=Bulmer-Thomas |first1=Ivor |title=Guldin's Theorem--Or Pappus's? |journal=Isis |date=1984 |volume=75 |issue=2 |pages=348–352 |doi=10.1086/353487 |jstor=231832 |url=https://www.jstor.org/stable/231832 |issn=0021-1753}}</ref>

==The first theorem==
The first theorem states that the [[surface area]] ''A'' of a [[surface of revolution]] generated by rotating a [[plane curve]] ''C'' about an [[axis of rotation|axis]] external to ''C'' and on the same plane is equal to the product of the [[arc length]] ''s'' of ''C'' and the distance ''d'' traveled by the [[Centroid|geometric centroid]] of ''C'':
<math display="block">A = sd.</math>

For example, the surface area of the [[torus]] with minor [[radius]] ''r'' and major radius ''R'' is
<math display="block">A = (2\pi r)(2\pi R) = 4\pi^2 R r.</math>

===Proof===

A curve given by the positive function <math> f(x) </math> is bounded by two points given by:

<math> a \geq 0 </math> and <math> b \geq a </math>

If <math> dL </math> is an infinitesimal line element tangent to the curve, the length of the curve is given by:

<math display="block"> L = \int_a^b dL = \int_a^b \sqrt{dx^2 + dy^2} = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>

The <math> y </math> component of the centroid of this curve is:

<math display="block"> \bar{y} = \frac{1}{L} \int_a^b y \, dL = \frac{1}{L} \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>

The area of the surface generated by rotating the curve around the x-axis is given by:

<math display="block"> A = 2 \pi \int_a^b y \, dL  = 2 \pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>

Using the last two equations to eliminate the integral we have:

<math display="block"> A = 2 \pi \bar{y} L </math>

==The second theorem==
The second theorem states that the [[volume]] ''V'' of a [[solid of revolution]] generated by rotating a [[plane figure]] ''F'' about an external axis is equal to the product of the area ''A'' of ''F'' and the distance ''d'' traveled by the geometric centroid of ''F''. (The centroid of ''F'' is usually different from the centroid of its boundary curve ''C''.) That is:
<math display="block">V = Ad.</math>

For example, the volume of the [[torus]] with minor radius ''r'' and major radius ''R'' is
<math display="block">V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2.</math>

This special case was derived by [[Johannes Kepler]] using infinitesimals.{{efn|Theorem XVIII of Kepler's ''Nova Stereometria Doliorum Vinariorum'' (1615):<ref>{{cite book |author-last=Kepler|author-first=Johannes |chapter=Nova Stereometria Doliorum Vinariorum | editor-last=Frisch|editor-first=Christian | title=Joannis Kepleri astronomi opera omnia | volume=4 | page=582 | location=Frankfurt |year=1870|orig-year=1615|access-date=2016-08-04 |publisher=Heyder and Zimmer |chapter-url=https://archive.org/details/joanniskeplerias04kepl}}</ref> "Omnis annulus sectionis circularis vel ellipticae est aequalis cylindro, cujus altitudo aequat longitudinem circumferentiae, quam centrum figurae circumductae descripsit, basis vero eadem est cum sectione annuli." Translation:<ref name="RdG2015" /> "Any ring whose cross-section is circular or elliptic is equal to a cylinder whose height equals the length of the circumference covered by the center of the figure during its circular movement, and whose base is equal to the section of the ring."}}

===Proof 1===

The area bounded by the two functions:

<math display="block"> y = f(x) , \, \qquad y \geq 0 </math>

<math display="block"> y = g(x) , \, \qquad f(x) \geq g(x) </math>

and bounded by the two lines:

<math> x = a \geq 0 </math> and <math> x = b \geq a </math>

is given by:

<math display="block"> A = \int_a^b dA  = \int_a^b [f(x) - g(x)] \, dx </math>

The <math> x </math> component of the centroid of this area is given by:

<math display="block"> \bar{x} = \frac{1}{A} \, \int_a^b x \, [f(x) - g(x)] \, dx </math>

If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by:

<math display="block"> V = 2 \pi \int_a^b x \, [f(x) - g(x)] \, dx </math>

Using the last two equations to eliminate the integral we have:

<math display="block"> V = 2 \pi \bar{x} A </math>

===Proof 2===
Let <math>A</math> be the area of <math>F</math>, <math>W</math> the solid of revolution of <math>F</math>, and <math>V</math> the volume of <math>W</math>. Suppose <math>F</math> starts in the <math>xz</math>-plane and rotates around the <math>z</math>-axis. The distance of the centroid of <math>F</math> from the <math>z</math>-axis is its <math>x</math>-coordinate
<math display="block">R = \frac{\int_F x\,dA}{A},</math>
and the theorem states that
<math display="block">V = Ad = A \cdot 2\pi R = 2\pi\int_F x\,dA.</math>

To show this, let <math>F</math> be in the ''xz''-plane, [[Parametric equation|parametrized]] by <math>\mathbf{\Phi}(u,v) = (x(u,v),0,z(u,v))</math> for  <math>(u,v)\in F^*</math>, a parameter region. Since <math>\boldsymbol{\Phi}</math> is essentially a mapping from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math>, the area of <math>F</math> is given by the [[Integration by substitution#Substitution for multiple variables|change of variables]] formula:
<math display="block">A = \int_F dA = \iint_{F^*} \left|\frac{\partial(x,z)}{\partial(u,v)}\right|\,du\,dv = \iint_{F^*} \left|\frac{\partial x}{\partial u} \frac{\partial z}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial z}{\partial u}\right|\,du\,dv,</math>
where <math>\left|\tfrac{\partial(x,z)}{\partial(u,v)}\right|</math> is the [[determinant]] of the [[Jacobian matrix and determinant|Jacobian matrix]] of the change of variables. 

The solid <math>W</math> has the [[torus|toroidal]] parametrization <math>\boldsymbol{\Phi}(u,v,\theta) = (x(u,v)\cos\theta,x(u,v)\sin\theta,z(u,v))</math> for <math>(u,v,\theta)</math> in the parameter region <math>W^* = F^*\times [0,2\pi]</math>; and its volume is
<math display="block">V = \int_W dV = \iiint_{W^*} \left|\frac{\partial(x,y,z)}{\partial(u,v,\theta)}\right|\,du\,dv\,d\theta.</math>

Expanding,
<math display="block">
\begin{align}
\left|\frac{\partial(x,y,z)}{\partial(u,v,\theta)}\right| & = 
\left|\det\begin{bmatrix}
\frac{\partial x}{\partial u}\cos\theta & \frac{\partial x}{\partial v}\cos\theta & -x\sin\theta \\[6pt]
\frac{\partial x}{\partial u}\sin\theta & \frac{\partial x}{\partial v}\sin\theta &  x\cos\theta \\[6pt]
\frac{\partial z}{\partial u}           & \frac{\partial z}{\partial v}           &  0
\end{bmatrix}\right| \\[5pt]
& = \left|-\frac{\partial z}{\partial v}\frac{\partial x}{\partial u}\,x + \frac{\partial z}{\partial u}\frac{\partial x}{\partial v}\,x\right|
=\ \left|-x\,\frac{\partial(x,z)}{\partial(u,v)}\right| = x\left|\frac{\partial(x,z)}{\partial(u,v)}\right|.
\end{align}
</math>

The last equality holds because the axis of rotation must be external to <math>F</math>, meaning <math>x \geq 0</math>. Now,
<math display="block">
\begin{align}
V &= \iiint_{W^*} \left|\frac{\partial(x,y,z)}{\partial(u,v,\theta)}\right|\,du\,dv\,d\theta \\[1ex]
&= \int_0^{2\pi}\!\!\!\!\iint_{F^*} x(u,v)\left|\frac{\partial(x,z)}{\partial(u,v)}\right| du\,dv\,d\theta \\[6pt]
& = 2\pi\iint_{F^*} x(u,v)\left|\frac{\partial(x,z)}{\partial(u,v)}\right|\,du\,dv \\[1ex]
&= 2\pi\int_F x\,dA
\end{align}
</math>
by change of variables.

==Generalizations==

The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions. 

Goodman & Goodman<ref name=generalizations>{{cite journal | last1=Goodman | first1=A. W. | last2=Goodman | first2=G. | title = Generalizations of the Theorems of Pappus | journal=The American Mathematical Monthly |volume=76 |issue=4 |pages=355–366 | jstor=2316426 | year = 1969 | doi = 10.1080/00029890.1969.12000217}}</ref> generalize the second theorem as follows. If the figure {{math|''F''}} moves through space so that it remains [[perpendicular]] to the curve {{math|''L''}} traced by the centroid of {{math|''F''}}, then it sweeps out a solid of volume {{math|1=''V'' = ''Ad''}}, where {{math|''A''}} is the area of {{math|''F''}} and {{math|''d''}} is the length of {{math|''L''}}. (This assumes the solid does not intersect itself.) In particular, {{math|''F''}} may rotate about its centroid during the motion. 

However, the corresponding generalization of the first theorem is only true if the curve {{math|''L''}} traced by the centroid lies in a plane perpendicular to the plane of {{math|''C''}}.

== In ''n''-dimensions ==
In general, one can generate an <math>n</math> dimensional solid by rotating an <math>n-p</math> dimensional solid <math>F</math> around a <math>p</math> dimensional sphere. This is called an <math>n</math>-solid of revolution of species <math>p</math>. Let the <math>p</math>-th centroid of <math>F</math> be defined by 

<math display="block">R = \frac{\iint_F x^p\,dA}{A},</math>

Then Pappus' theorems generalize to:<ref>{{Cite book |url=https://cds.cern.ch/record/254647 |title=An introduction to the geometry of n dimensions|last=McLaren-Young-Sommerville |first=Duncan |date=1958 |publisher=Dover |location=New York, NY |chapter=8.17 Extensions of Pappus' Theorem}}</ref>

<blockquote>

Volume of <math>n</math>-solid of revolution of species <math>p</math> 
<br>  =  (Volume of generating <math>(n{-}p)</math>-solid) <math>\times</math> (Surface area of <math>p</math>-sphere traced by the <math>p</math>-th centroid of the generating solid) 

</blockquote>

and

<blockquote>

Surface area of <math>n</math>-solid of revolution of species <math>p</math>
<br>  = (Surface area of generating <math>(n{-}p)</math>-solid) <math>\times</math> (Surface area of <math>p</math>-sphere traced by the <math>p</math>-th centroid of the generating solid) 

</blockquote>

The original theorems are the case with <math>n=3,\, p = 1</math>.

== Footnotes ==
{{notelist}}

== References ==
{{reflist}}

==External links==
{{commons category|Pappus-Guldinus theorem}}
*{{MathWorld|title=Pappus's Centroid Theorem|urlname=PappussCentroidTheorem}}

[[Category:Theorems in calculus]]
[[Category:Geometric centers]]
[[Category:Theorems in geometry]]
[[Category:Area]]
[[Category:Volume]]
[[Category:Greek mathematics]]