When do we use $2\pi $ vs using $\pi $ in finding the volume of a region?

I've seen that sometimes finding the integral volume of a rotated shape in the xy-plane is multiplied by $2\pi $, and other times it is only multiplied by $\pi $. Can someone please tell me the difference?

I've seen that sometimes finding the integral volume of a rotated shape in the xy-plane is multiplied by $2\pi $, and other times it is only multiplied by $\pi $. Can someone please tell me the difference?