How would I find the volume of the body formed by revolving the circle $r = f(\theta) = \cos\theta$ about the line $\theta = \frac{\pi}{2}$ ?
(This is the circle of radius $1$ centered at $(0,1)$ which generates a torus)
Do I integrate to find the area of the circle and then plug that value into another integration equation to find the volume? If so, how would this be done?
For any set $G\subseteq \mathbb R^n$, the volume of $G$ is simply $$\int_G 1 d\vec x.$$
This means that the volume of your torus is $$\int \int \int_G 1 dxdydz.$$ Converting the coordinates to a polar form ($(x,y,z)\to (r,\phi,z)$) will allow you to set the bounds of $r, \phi$ and $z$ much more easily than if you are doing it in the cartesian coordinates, but still, there is no avoiding some amount of work.