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Volume of a Cylinder

Date: 09/05/97 at 16:00:04
From: Lee Adams
Subject: Volume of a cylinder

I would like to know how to calculate the volume of a cylinder laid on 
it major axis if I know the height of the top of the section and the 
height of the bottom of the section. I can calculate the volume if the 
section is from the bottom to part way up, etc., but cannot calculate 
it if the section is like this:

         /        \
        /          \ 
     __|____________|__  to here
     __|XXXXXXXXXXXX|__  from here
        \          /
         \        /

Can you please show me how to do it? Or does it require calculus?

Date: 09/11/97 at 16:30:34
From: Doctor Rob
Subject: Re: Volume of a cylinder

If you can do it from the bottom up, then the answer you seek is just
the difference of the volume from the bottom to the upper line less 
the volume from the bottom to the lower line.

The volume is the length of the cylinder times the cross-sectional 
area. In your case, you have the area of a circle less the area of 
two segments of the circle.

The area of the circle is Pi*r^2, where r is the radius. Suppose the
distance from the bottom of the circle to the lower line is A, and 
the distance from the top of the circle to the upper line is B.  
Then the area of the X-ed region above is:

  Pi*r^2 - [Pi*r^2/2 - (r-A)*Sqrt[2*A*r - A^2] - r^2*Arcsin[(r-A)/r]]
         - [Pi*r^2/2 - (r-B)*Sqrt[2*B*r - A^2] - r^2*Arcsin[(r-B)/r]]
  = (r-A)*Sqrt[2*A*r - A^2] + r^2*Arcsin[(r-A)/r] +
    (r-B)*Sqrt[2*B*r - A^2] + r^2*Arcsin[(r-B)/r]

The arcsine results must both lie in [0, Pi/2] provided both A <= r
and B <= r.  If one or the other is greater than r, a similar but
different formula will apply.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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