The math behind mass properties

I am in the process of writing either a macro or vb standalone to design
a part one of whose design criteria is volume.
The part is basically a revolve. It consists of 6 design criteria
dimensions + design volume. That leaves one dimension as the only variable.
I have been estimating the initial variable, checking mass properties,
adjusting, rebuilding, rechecking volume, ect until I get it close enough.
The part revolve is made up of a series of tangent arcs and lines so
doing the math by hand would take much longer than guessing until correct so
I would like to macro the funtion as I do it quite often ( alter cumbustion
chamber volume to adjust for desired compression ratio ).
The issue that I am having is that I can't seem to find any sites/books
that have the formulae that I'm after. Is anyone able to direct me to a
source?

 

Several years ago there was a scuba tank model that allowed you to specify
some parameters and then let Excel solver determine the optimal size to
maximize the volume while minimizing the weight. I think this is really
what would do what you want to do. Unfortunately, I don't have a copy -
anyone else?

WT

 

What you probably want is the volume of a solid of revolution. There
are two main ways to find this using integrals, both are explained at
http://mathworld.wolfram.com/SolidofRevolution.html

If you do a search on "volume solid revolution", you'll find more
sites. The basic thing is knowing a function of the shell. You will
probably need to apply some numeric integration method if you can't
come up with an antiderivative of your curve. Look for information on
Simpson's rule, or the trapezoid rule. Here's a page with some links
to more information about numeric integration methods:
http://mathworld.wolfram.com/NumericalIntegration.html

If you need more help, if I could know the curve that you're revolving,
I could probably give you a formula.

Hope that helps,
Jonathan Anderson

 

Those are the formulae that I was after. I now recall seeing them about 10
years ago in book form, and being very intimidated. Thanks for the links.
Maybe this time I'll attempt to tackle the problem.