Heptagonal number: Difference between revisions

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→‎Test for heptagonal numbers: what? "Let y be a positive integer", and then give an equation in terms of n? This needs to be reworded: now it is really not helping anything.
 
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In mathematics, '''Pappus' centroid theorem''' (also known as  the '''Guldinus theorem''', '''Pappus–Guldinus theorem''' or '''Pappus' 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.
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The theorem is attributed to [[Pappus of Alexandria]] and [[Paul Guldin]].
 
==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 its [[Centroid|geometric centroid]].
 
: <math>A = sd.\,</math>
 
For example, the surface area of the [[torus]] with minor [[radius]] ''r'' and major radius ''R'' is  
 
: <math>A = (2\pi r)(2\pi R) = 4\pi^2 R r.\,</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 its geometric centroid.
 
: <math>V = Ad.\,</math>
 
For example, the volume of the [[torus]] with minor radius ''r'' and major radius ''R'' is
 
: <math>V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2.\,</math>
 
==External links==
*{{MathWorld|title=Pappus's Centroid Theorem|urlname=PappussCentroidTheorem}}
 
[[Category:Theorems in calculus]]
[[Category:Geometric centers]]
[[Category:Theorems in geometry]]

Latest revision as of 15:23, 22 June 2014

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