# 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 $a\le x\le b$, the area of the formed surface of revolution^{} is

$$A=\mathrm{\hspace{0.33em}2}\pi {\int}_{a}^{b}y(x)\sqrt{1+{[{y}^{\prime}(x)]}^{2}}\mathit{d}x.$$ |

This can be concisely written

$A=\mathrm{\hspace{0.33em}2}\pi {\displaystyle {\int}_{s}}y\mathit{d}s$ | (1) |

since differential-geometrically, the product $\sqrt{1+{[{y}^{\prime}(x)]}^{2}}dx$ is the arc-element. We rewrite (1) as

$$A=s\cdot 2\pi \cdot \frac{1}{s}{\int}_{s}y\mathit{d}s.$$ |

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

$$A=s\cdot 2\pi R$$ |

which states the first Pappus’s centroid theorem.

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

$$a\le x\le b,0\le {y}_{1}(x)\le y\le {y}_{2}(x),$$ |

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 $y={y}_{1}(x)$ and $y={y}_{2}(x)$ is $[{y}_{2}(x)+{y}_{1}(x)]/2$ when the abscissa is $x$; the area of this element with the width $dx$ is $[{y}_{2}(x)-{y}_{1}(x)]dx$. Thus we get the equation

$$R=\frac{1}{A}{\int}_{a}^{b}\frac{{y}_{2}(x)+{y}_{1}(x)}{2}[{y}_{2}(x)-{y}_{1}(x)]\mathit{d}x$$ |

which may be written shortly

$R={\displaystyle \frac{1}{2A}}{\displaystyle {\int}_{a}^{b}}({y}_{2}^{2}-{y}_{1}^{2})\mathit{d}x.$ | (2) |

The volume of the solid of revolution is

$$V=\pi {\int}_{a}^{b}({y}_{2}^{2}-{y}_{1}^{2})\mathit{d}x=A\cdot 2\pi \cdot \frac{1}{2A}{\int}_{a}^{b}({y}_{2}^{2}-{y}_{1}^{2})\mathit{d}x.$$ |

By (2), this attains the form

$$V=A\cdot 2\pi R.$$ |

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 |