Finding The Volume Enclosed by a Torus

In summary, the goal is to find the volume enclosed by the torus rho = sin theta. The attempt at a solution involved setting the limits of phi from 0 to pi, theta from 0 to 2 pi, and rho from 0 to sin theta. However, this resulted in a volume of 0. It was then discussed that there are different conventions for spherical coordinates and it was determined that theta is the azimuthal angle in this case. The Pappus theorem was also mentioned as a method for finding volumes of rotation.
  • #1
sozo91
5
0

Homework Statement



Find the volume enclosed by the torus rho = sin theta.


Homework Equations





The Attempt at a Solution



I tried setting the limits as phi from 0 to pi, theta from 0 to 2 pi, and rho from 0 to sin theta. However, if i do that, i get a volume of 0. How should i set up the limits?
 
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  • #2
Which angle is theta? There is more than one convention for spherical coordinates. From your limits I would guess its the azimuthal angle, but from rho=sin(theta) I'd guess its the polar angle. Can you show the integral you finally got?
 
  • #3
I'm an idiot. It's the azimuthal angle. I tried solving it as the polar angle. Thanks.
 
  • #4
Solutions for volumes of rotation are easy using the Pappus theorem. This is attributed to Pappus of Alexandria, but was first proved by the Swiss mathematician Guldin. The theorem states that the volume is the area of the profile times the distance that the center of gravity of the profile moves. The axis of rotation cannot pass through the profile.
 

1. What is a torus?

A torus is a three-dimensional geometric shape characterized by a circular cross-section and a hole in the center. It resembles a donut or an inner tube.

2. How is the volume of a torus calculated?

The volume of a torus can be calculated by using the formula V = π²r²h, where "r" is the radius of the circular cross-section and "h" is the distance from the center of the torus to the center of the circular cross-section.

3. Can the volume of a torus be found using calculus?

Yes, the volume of a torus can also be calculated by using calculus, specifically the integral of a function that represents the cross-sectional area of the torus.

4. What are some real-life applications of finding the volume enclosed by a torus?

Finding the volume of a torus is useful in various fields such as engineering, architecture, and physics. It can help determine the capacity of pipes, calculate the amount of paint needed to cover a torus-shaped object, or estimate the volume of blood flow in a blood vessel.

5. Are there any other methods for finding the volume of a torus?

Yes, there are other methods for finding the volume of a torus, such as using geometric constructions or using specialized software programs. However, the most common and widely-used method is the formula V = π²r²h.

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