I'm taking Calculus 2 and I have this question:
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?
Whether to use $\pi$ or $2\pi$ depends on the choice of the two common integration methods.
1) If the disk-of-circles method is used, disk area $\pi r^2$ is applied. The integration is then over a stack of disks,
$$V = \int_a^b \pi x^2 dx$$
2) If the shell-of-cylinders method is used, cylinder circumference $2\pi r$ is applied. The integration is then through cylinder walls,
$$V=\int_a^b 2\pi xy(x)dx$$