After playing around with 3D parametric equations on my calculator (modifying the equations of a standard torus), I came across a shape that I like. The equations are:
$$x=(2+\sin t)\cos u$$ $$y=(2+\cos t)\sin u$$ $$z=\sin t\cos t$$
I would like to know how to find the volume and surface area of the shape. I have read a bit about surface integrals and Jacobian transformations to find the volume of a torus. http://mathworld.wolfram.com/Torus.html
I went through an example of finding the volume of a normal (donut) torus with the jacobian transformation method i.e. $\frac{\partial(x,y,z)}{\partial(u,v,w)}$ and in the end the answer simplifies to $(c + a\cos v) a$.
I would very much appreciate if someone could help me with this and explain how all this remapping and multivariable changes work.