From: ken@straton.demon.co.uk (Ken Starks)
Newsgroups: sci.math
Subject: Re: @@@ WEIRD VOLUME PROOF
Date: Mon, 23 Nov 1998 01:36:30 GMT
"Robert \"Tralfaz\" Armagost" wrote:
> Use Archimides' method of equilibrium (basicly a geometric form of
> integration without the rigorous use of limits) If you already know the
> formula, his method of exhaustion also works quite well.
I have a vague feeling that these methods of Archimedes actually
followed, and possibly depended upon, the result we are trying to prove.
This would make them invalid. But if you don't care about such
nicities, you could use one of Pappus' theorems, which are among
my favorite of these pre-calculus methods.
Volume of Solid of Revolution =
Area of Region used to sweep out solid
* Distance moved by Centroid of that Region
For a Right-angled triangle with legs h and r, we get a cone.
Centriod at 1/3rd of way out ( distance of r/3 from axis )
Area = (h * r )/2
Distance = 2 * Pi * r/3
=> Volume = Pi * h * r * r/3 as required.
Ken, __O
_-\<,_
(_)/ (_)
Virtuale Saluton.
==============================================================================n
From: kunkel@REMOVEcnw.com (Paul Kunkel)
Newsgroups: sci.math
Subject: Re: Pappas
Date: Fri, 4 Dec 1998 14:11:30 -0800
In article <3668517E.3EE2AC75@wpi.edu>, mkcheme@wpi.edu (Mike Kuczewski)
says...
> I'm a freshman at WPI in CalcIV doing a project onm centers of mass.
> Part of it onvolves using Pappas's Theorem, and i can't quite remember
> how it goes. Can somebody here tell me what it is/where I can find it?
I am going to quote the theorem from _Calculus With Analytic
Geometry_ 3rd edition, by Edwin J. Purcell, 1978 Prentice-Hall. You
might have trouble finding this particular book though. If they have a
math library over there at the Wyoming Petrography Institute, then I
would suggest going to the biggest fattest calculus books and browsing
through the indices. Look for a book with good illustrations. That can
make it easier to get the gist of the theorem.
Kunkel
Pappus's Theorem:
Let f and g be continuous on [a,b], with g(x) <= f(x) for a <= x <= b.
The volume of the solid of revolution generated by revolving the region
R = {P:(x,y)|a <= x <= b, g(x) <= y <= f(x)} about a horizontal or
vertical line, in the plane of R but not intersecting R, is equal to the
area of R times the circumference of the circle described by the centroid
of R.
==============================================================================
From: zaccaria3@msn.com
Newsgroups: sci.math
Subject: Re: Pappas
Date: Mon, 07 Dec 1998 06:24:29 GMT
In article <3668517E.3EE2AC75@wpi.edu>,
mkcheme@wpi.edu wrote:
> I'm a freshman at WPI in CalcIV doing a project onm centers of mass.
> Part of it onvolves using Pappas's Theorem, and i can't quite remember
> how it goes. Can somebody here tell me what it is/where I can find it?
> -Mike
>
Mike,
Get hold of frank ayres, jr, PROJECTIVE GEOMETRY, in the Schaum Outline
Series, and check out pages 26 and 28. My copy came out in 1967, and the
Schaum Co. has been bought, but you can find a copy in a good library.
RYZ
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==============================================================================
From: Floor van Lamoen
Newsgroups: sci.math
Subject: Re: Pappas
Date: Mon, 07 Dec 1998 16:05:35 +0100
Mike Kuczewski wrote:
>
> I'm a freshman at WPI in CalcIV doing a project onm centers of mass.
> Part of it onvolves using Pappas's Theorem, and i can't quite remember
> how it goes. Can somebody here tell me what it is/where I can find it?
> -Mike
Maybe that's becuase it's Pappos' or Pappus' theorem:
When points A1,A3 and A5 are on line l and points A2, A4 and A6 are on
line m then A1A2/\A4A5 (/\:intersection) A1A6/\A3A4 and A2A3/\A5A6 are
collinear.
Best regards,
Floor van Lamoen.
==============================================================================
From: ken@straton.demon.co.uk (Ken Starks)
Newsgroups: sci.math
Subject: Re: Pappas
Date: Mon, 07 Dec 1998 19:24:27 GMT
Mike Kuczewski wrote:
>> I'm a freshman at WPI in CalcIV doing a project onm centers of mass.
>> Part of it onvolves using Pappas's Theorem, and i can't quite remember
>> how it goes. Can somebody here tell me what it is/where I can find it?
Pappus had quite a few theorems, but there are two main ones, related to a
surface or solid of revolution.
Under certain conditions:
1. Scenario:
Surface S generated by rotating a curve C with centre of mass P.
Result:
Surface area of S = Length of C * distance moved by P
2. Scenario:
Solid V generated by rotating plane figure F with centre of mass at P
Result:
Volume of V = Area of F * distance moved by P
'Centre of mass' defined as if there is an even linear density or area
density.
Note: square a few things and use Pythagoras' theorem to get a useful
result about the moment of inertia.
Ken, __O
_-\<,_
(_)/ (_)
Virtuale Saluton.