Pappus's centroid theorem

File:Pappus centroid theorem areas.gif

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 theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of AlexandriaTemplate:Efn and Paul Guldin.Template:Efn 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>Template:Cite journal</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 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 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.Template:Efn

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, 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 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 of the change of variables.

The solid <math>W</math> has the 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>Template:Cite journal</ref> generalize the second theorem as follows. If the figure Template:Math moves through space so that it remains perpendicular to the curve Template:Math traced by the centroid of Template:Math, then it sweeps out a solid of volume Template:Math, where Template:Math is the area of Template:Math and Template:Math is the length of Template:Math. (This assumes the solid does not intersect itself.) In particular, Template:Math may rotate about its centroid during the motion.

However, the corresponding generalization of the first theorem is only true if the curve Template:Math traced by the centroid lies in a plane perpendicular to the plane of Template:Math.

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>Template:Cite book</ref>

Volume of <math>n</math>-solid of revolution of species <math>p</math>
= (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)

and

Surface area of <math>n</math>-solid of revolution of species <math>p</math>
= (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)

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

External links

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