To: "Richard Schwartz" 
From: MLThiebaux 
Cc: atm{at}shore.net
Reply-To: MLThiebaux 


Hello Richard --

Simpson's rule is exact for integrating polynomials of up to degree 3.  The
crossectional area for a paraboloid is 2*pi*R*y while for a sphere it is
pi*(2*R*y-y^2), polynomials of degree 1 and 2, respectively. (y is the
height of the surface above the bottom tangent plane).  So using Simpson's
rule for integrating over y from 0 to h with just 2 intervals as you did is
exact for either curve.

BTW, that last formula quoted below is for a spherical SECTOR, not the
spherical SEGMENT that we want.

Martial Thiebaux
Rawdon Hills, Nova Scotia


Richard Schwartz wrote:

...
>3.  Numerical Integration: use numerical integration such as simpson's rule
>to find the volume.   This is actually similar to the calculus method
>because numerical integration assumes some kind of polynomial for the
>integrand.   One of my favorite methods is based on this.   It is exact for
>the volume of a LOT of things, and close for most everything else.
>
...
>
>Then there is the distasteful fourth way:  look it up in a standard
>handbook.  There I found that the volume is...
>
>(2/3)*pi*h*R^2, where h is the saggita  (which you know how to compute if
>your are a telescope person)
...
>
>. . . Richard
>
>

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