Editing Pappus's centroid theorem (section)

== 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>.