Calculating Volume with Integration: 4/3(pi)r^3

In summary, the conversation revolves around finding the volume of a sphere using different techniques, such as integration and differentiation. The participants also discuss the formula for the surface area of a sphere and its relation to the volume of a cone. One participant realizes they were mistaken about the formula for the surface area of a sphere. Ultimately, the conversation ends with a realization about the volume of a cone and its comparison to the volume of a sphere.
  • #1
Silverious
52
0
If I intergrate 2(pi)r I get (pi)r^2

If I integrate that I get 1/3 (pi)r^3, which is close to the volume of a sphere. But where do I get a 4/3(pi)r^3 ?
 
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  • #2
Differentiate the volume expression, and find a suitable interpretation of the result.
 
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  • #3
beacause disks are not the infinitesimal shells that add up to a solid sphere
 
  • #4
Would it work if I used half of the area forumula, and used the disk method to rotate it about an axis? Should that give me 4/3(pi)r^3?

I would find out for myself but I'm a little busy.

Thanks for the replies.
 
  • #5
that's a different technique: add up the shells of surface area of a sphere
 
  • #6
OKay I'm starting to understand.

So I use the surface area formula(for a sphere). Integrate from 0 to r? :uhh:

Edit: Oh my god. I feel so stupid...

You see, I didn't know the formula for the surface area of a sphere. ...

Oh well. Anyways, uum. So then, what is 1/3(pi)r^3? Without looking it up, making a wild guess...is it the volume of a cylinder?

Edit2: I really need to stop thinking... I have no clue what I'm talking about. :yuck: So is 1/3(pi)r^3 just nonsense?

Edit3: Interesting that a cone's volume is 1/3(pi)r^2 h. Good I have much time to think about it.
 
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  • #7
the surface area of a sphere is 4(pi)r^2
 

1. What is the formula for calculating volume using integration?

The formula for calculating volume using integration is 4/3(pi)r^3, where r is the radius of the object.

2. How is integration used to calculate volume?

Integration is used to find the volume of irregularly shaped objects by dividing it into infinitesimally small slices, calculating the volume of each slice, and then adding them together using integration.

3. Can this formula be applied to any shape?

No, this formula is only applicable to objects with a spherical shape. Other shapes may require different formulas or methods of calculation.

4. What units should be used for the radius?

The radius should be measured in the same units as the other dimensions of the object. For example, if the length and width are measured in meters, the radius should also be measured in meters.

5. How accurate is the volume calculated using integration?

The accuracy of the calculated volume depends on the accuracy of the measurements used to determine the radius. The smaller the increments used in the integration process, the more accurate the result will be.

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