Advice on volume of solids NOT of revolution?

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SUMMARY

This discussion focuses on strategies for calculating the volumes of solids that are not solids of revolution. Participants emphasize the importance of understanding the cross-sectional area of the solid, which varies based on the specific shape being analyzed. For example, when calculating the volume of a cylinder with a hole bored through it, the integral setup requires knowledge of the cross-sectional function. The conversation also references Cavalieri's principle as a useful concept for volume calculation.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with cross-sectional area concepts
  • Knowledge of Cavalieri's principle
  • Basic geometry of three-dimensional shapes
NEXT STEPS
  • Research the application of Cavalieri's principle in volume calculations
  • Explore methods for calculating volumes of irregular solids
  • Study the setup of integrals for various three-dimensional shapes
  • Learn about different cross-sectional area formulas for common solids
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and geometry, as well as professionals involved in engineering and design who require volume calculations for non-revolution solids.

Gauss177
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Anyone have any advice for finding volumes of solids that are not solids of revolution? I have a much more difficult time starting these kinds of problems compared to revolving ones.
 
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Gauss177 said:
Anyone have any advice for finding volumes of solids that are not solids of revolution? I have a much more difficult time starting these kinds of problems compared to revolving ones.

I'm not sure anyone can help you unless you become a bit more specific.
 
I mean general things to do or whatever to start these kinds of problems. And also what would a typical cross section be like? Whereas when using washer method/shell method the solid is revolved so cross sections are circular, do the cross sections for these problems depend entirely on the question?

for example:
A hole of radius r is bored through a cylinder of radius R > r at right angles to the axis of the cylinder. Set up an integral (no need to evaluate) for the volume cut out.
 
If you know the function which defines the cross section, then you can calculate the integral. You may want to do some google-ing on Cavalieri solids. I hope I was at least a bit helpful. :wink:
 
say you are integrating a cube, then the area of a typical cross section would be l*w. for a cone it is pi*r^2 etc..it really depends on what kind of solids you are trying to find the volume of. But a cross section is just an infinitely thin slice out of the solid.
 
There is no general method for finding the volume of an arbitrary 3 dimensional solid.
 
Yes there is, putting it in water :D
 

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