- #1
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.
Advice on volume of solids NOT of revolution?
- Thread starter Gauss177
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- Revolution Solids Volume
In summary, the conversation discusses the difficulty in finding the volumes of solids that are not solids of revolution. The question asks for advice on how to approach these problems and what a typical cross section would look like. It is mentioned that knowing the function defining the cross section can help with calculating the integral. Additionally, the concept of Cavalieri solids is brought up and it is mentioned that there is no general method for finding the volume of an arbitrary 3-dimensional solid. The conversation ends with a humorous suggestion of using water to find the volume.
- #2
radou
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Gauss177 said:
I'm not sure anyone can help you unless you become a bit more specific.
- #3
Gauss177
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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.
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.
- #4
radou
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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. 
- #5
trajan22
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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.
- #6
HallsofIvy
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There is no general method for finding the volume of an arbitrary 3 dimensional solid.
- #7
Gib Z
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Yes there is, putting it in water :D
1. How do you calculate the volume of a solid object?
To calculate the volume of a solid object, you need to know its dimensions (length, width, and height). The formula for volume is V = length x width x height. Make sure to use consistent units of measurement.
2. What is the difference between the volume of a solid and the volume of a solid of revolution?
The volume of a solid is the amount of space it occupies, while the volume of a solid of revolution is the space created by rotating a 2-dimensional shape around an axis. The volume of a solid of revolution is typically calculated using the method of cylindrical shells or the method of discs/washers.
3. Can the volume of a solid of revolution be negative?
No, the volume of a solid of revolution cannot be negative. It represents the physical space occupied by the solid, so it must be a positive value.
4. Is there a general formula for finding the volume of a solid of revolution?
Yes, there are two general formulas for finding the volume of a solid of revolution: the method of cylindrical shells and the method of discs/washers. The choice of which formula to use depends on the shape of the 2-dimensional cross section being rotated.
5. What are some real-life applications of calculating volumes of solids of revolution?
Calculating volumes of solids of revolution is commonly used in engineering and design, such as in the production of cylindrical objects like pipes and cans. It is also used in fields like architecture, where curved structures like domes and arches are involved. Additionally, understanding volumes of solids of revolution is important in physics and calculus for solving real-world problems involving rotational motion.
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