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Re: Trouble calculating a volume
*To*: mathgroup at smc.vnet.net
*Subject*: [mg33840] Re: [mg33806] Trouble calculating a volume
*From*: Rob Pratt <rpratt at email.unc.edu>
*Date*: Fri, 19 Apr 2002 02:28:56 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
On Tue, 16 Apr 2002, Jay D. Martin wrote:
> Respected Colleagues:
>
> I am attempting to calculate the volume of the frustum of a cone.
> I have been able to calculate the volume of a cone using a triple
> integral as follows:
>
> Integrate[1, {x, 0, h}, {y, (-r)*(x/h), r*(x/h)},
> {z, -Sqrt[(r*(x/h))^2 - y^2], Sqrt[(r*(x/h))^2 - y^2]}]
>
> This integral does evaluate to the correct answer
>
> (1/3)*h*Pi*r^2
>
> where r is the base diameter and h is the height.
>
> In order to calculate the volume of the frustum of a cone, I have used the
> following equation :
>
> Simplify[Integrate[1, {x, 0, h},
> {y, -(r1 + ((r2 - r1)*x)/h), r1 + ((r2 - r1)*x)/h},
> {z, -Sqrt[(r1 + ((r2 - r1)*x)/h)^2 - y^2],
> Sqrt[(r1 + ((r2 - r1)*x)/h)^2 - y^2]}]]
>
>
> where r1 and r2 are the top and bottom radii of the section and h
> is the height. This integral evaluates to include Logarithms and
> imaginary numbers
>
> (I*Sqrt[h^2*r2]*(r1^2 + r1*r2 + r2^2)*
> (Log[-2*I*h*r2] - Log[2*I*h*r2]))/(3*Sqrt[r2])
>
> when the solution should be
>
> (1/3)*h*Pi*(r1^2+r1*r2+r2^2)
>
> Which, if you substitute r1=0 and r2=r you get the volume of
> a cone.
>
> I am using Mathematica Version 4.1.2.0
> running on Windows 2000
>
> Thank you for any help
>
> Jay Martin
>
> PS: what I am ultimately after are the moments of inertia of the section
> That is why I am using the triple integral in terms of x, y, and z
Using Simplify with assumptions {h >= 0, r2 > 0} yields your desired
answer.
Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill
rpratt at email.unc.edu
http://www.unc.edu/~rpratt/
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