Editing Pappus's centroid theorem (section)

==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.