Rotating a circle around the y-axis

Question

What is the volume obtained when a circle of radius 2 with center (2,0) is rotated about the y-axis? I keep getting 0, but that can't be right!

Answer 1

The points on the right half of the circle are at distance $x=f(y)=2+\sqrt{4-y^2}$ from the $y$-axis, and the points on the left half are at distance $x=g(y)=2-\sqrt{4-y^2}$ from the $y$-axis.

The volume you are trying to calculate is that of the body you get by revolving $x=f(y)$, $y\in[-2,2]$, about the $y$-axis of which you need to subtract the volume of the body gotten by revolving $x=g(y)$,$y\in[-2,2]$ about the $y$-axis. Do you see why?

Answer 2

Try the second Pappus's centroid theorem then.

Answer 3

This can very easily solved by using simple geometry as follows

In general, if a circle, having a radius $r$, is rotated about one of its tangents then the solid obtained is a $\color{blue}{\text{horn torus}}$ with radius $r$ & mean radius $r_m=r$ measured from the center of circle.

The volume of the horn torus is given as $$V_{\text{horn torus}}=(\pi r^2)(2\pi r)=2\pi^2r^3$$

In this case, the circle of radius $r=2$ is rotated about the y-axis (i.e. tangent) hence the volume of the solid obtained i.e. horn torus is $$2\pi^2(2^3)=16\pi^2$$$$\bbox[5px, border:2px solid #C0A000]{\color{red}{\text{volume}=16\pi^2\approx 157.9136704\ \text{cubic units}}}$$