Question about rotating a circle around the y axis to form a torus

Question

Can someone briefly describe the steps I could take to solve a problem in which I form a torus by rotating a circle around the y-axis? An example question is "what is the volume of the shape when (x^2+y^2=1) is rotated about the line y=2?"

Answer 1

Briefly, you take the set $S$ defined in th plane $\mathbb R^2$, and in order to compute any quantity involved to the rotation of $S$ around an axis, you need to establish an equation of the resulting set (obtained when rotating every point in $S$), or if you want to compute the volume, establish the volume infinitesimal element in appropiate coordinates.

In the example you gave, it is natural to parameterize the resulting set with toroidal coordinates centered in $(0,2)$, i.e. the vector joigning $(0,2)$ with the center of a circle, and then the vector joigning the center of a circle to any point inside of it.