Can we derive a formula for the volume of revolution in polar coordinates?
My attempt was so consider a very small piece of a circle that is bounded by the counterclockwise angles $\theta$ and $\theta+d \theta$ in the $xy$ plane. Then the lines forming those angles are $y=(\tan \theta) x$ and $y=\tan (\theta+d \theta) x$. So I suppose to get the (approximate) volume obtained by rotating this little about the $x$ axis. We should consider,
$$\int\limits_{0}^{r \cos \theta} (\tan (\theta+d \theta) x)^2-(\tan (\theta) x)^2 dx$$
This is a bit sloppy and even so, I'm not sure where to go from here.