Finding The Volume Enclosed by a Torus

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    Torus Volume
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Homework Help Overview

The problem involves finding the volume enclosed by a torus defined by the equation ρ = sin(θ), with a focus on the appropriate limits for integration in spherical coordinates.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set integration limits for φ, θ, and ρ but encounters a volume of 0, prompting questions about the correct setup. Some participants question the definition of the angle θ in the context of spherical coordinates, suggesting a potential misunderstanding of its role.

Discussion Status

Participants are exploring different interpretations of the angle θ and its implications for the integration limits. One participant acknowledges a mistake in identifying θ as the polar angle rather than the azimuthal angle, indicating a productive clarification in the discussion.

Contextual Notes

There is a mention of multiple conventions for spherical coordinates, which may affect the setup of the problem. The original poster's confusion about the limits suggests a need for clearer definitions and assumptions regarding the coordinate system being used.

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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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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?
 
I'm an idiot. It's the azimuthal angle. I tried solving it as the polar angle. Thanks.
 
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.
 

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