1. ## Spherical Parametrization for volume calculation - Your help needed.

Hi - I'm stuck on a problem to find the volume of a Tori/Torus. I can find a 2 variable parameterization (which allows me to compute the surface area by double integral for the first part of the question) but cant find a way to set it in 3 (spherical coords) to compute the volume.

The equation of the shape is:

$y^2+(3-\sqrt{x^2+z^2})^2 = 4$

If anyone can help me out with the transformation to spherical and intervals I'd appreciate it...

Thanks, F

2. ## Re: Spherical Parametrization for volume calculation - Your help needed.

A parameterization for the torus is given by
$(x,y,z)\to ((R+\rho \cos \phi ) \cos t , (R+\rho \cos( \phi)) \sin t , \rho \sin \phi)$, where $\phi \in [0,2Pi] , t\in [0,2Pi] , \rho \in [-s,s]$.

Here R is the major radius and s is the minor radius. Basically you can construct this parameterization by beginning with the large circle $(R \cos t , R \sin t, 0)$.
This parameterization can be extended to the torus by adding smaller circles of radius s at each point.

By the way, you can calculate the volume of a torus simply by transforming it into a cylinder. Given a major radius R its height is then 2 pi R and the base has a surface of pi s^2 where s is the minor radius.