Prove volume of a torus equation

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    Torus Volume
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SUMMARY

The volume of a torus can be calculated using integration techniques, specifically through the method of cylindrical shells. This involves slicing the torus into horizontal rings or vertical discs and integrating to find the volume. The discussion highlights that any cross-sectional shape, including polygons or ellipses, can be revolved around a circle to derive the volume. Pappus' centroid theorem is also suggested as a potential method for calculating the volume of revolution.

PREREQUISITES
  • Understanding of integration techniques in calculus
  • Familiarity with the method of cylindrical shells
  • Knowledge of Pappus' centroid theorem
  • Basic concepts of volume of revolution
NEXT STEPS
  • Study the method of cylindrical shells for volume calculation
  • Explore Pappus' centroid theorem and its applications
  • Learn about different cross-sectional shapes and their revolutions
  • Practice integration techniques for calculating volumes of solids of revolution
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and geometry, as well as anyone interested in understanding the volume calculations of three-dimensional shapes like a torus.

brandy
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how does the formula for volume of a torus work.
is there a proof with integration??

could you use an ellipse?
 
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It's a surface of revolution, so yes, there is a formula (method of cylindrical shells).
 
brandy said:
how does the formula for volume of a torus work.
is there a proof with integration??

Hi brandy! :smile:

Yes, slice it into horizontal rings (or vertical discs), and integrate. :wink:
could you use an ellipse?

uhhh? :confused:
 
by ellipse i mean, an ellipse is revolved around a circular ring.
sort of like taking a torus and stretching it upward...
get me?
 
also what could you use a shape like a polygon or something and revolve it around a circle.

how could you would out the volume?
 
brandy said:
by ellipse i mean, an ellipse is revolved around a circular ring.
sort of like taking a torus and stretching it upward...
get me?
brandy said:
also what could you use a shape like a polygon or something and revolve it around a circle.

how could you would out the volume?

ah! got you! :smile:

yes, you can use any cross-section shape …

if you slice it into horizontal rings, you just need to know the width of the polygon at each height …

and if you use vertical discs, you just need to know the height of the polygon at each width :wink:
 
sorry. i kept rewording what i was going to say and i didnt read what i wrote.
so all you need is the x,y positions ?
can you explain how this works??
keep in mind i know nothing AT ALL.
 
i mean like wat do u do to the points... to get the volume of revolution
 
Have you considered using Pappus' centroid theorem?
 

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