Rotating an interval about the origin? 
Question: Working in the $(x,y)$ plane I write an interval on the $x-$axis, $I=[n-1,n].$ How do I construct an integral to compute
  the area by rotating $I$ about the origin $(0,0) ?$ Rotating $I$
  results in an annulus I am interested in the construction of the integral not in the explicit formula for the area of the annulus


Rotating the interval by the angle $\theta_0$ yields the following integral
$$\int_0^{\theta_0} \int_{n-1}^n r\,dr\,d\theta = \int_0^{\theta_0} \frac{n^2 - (n-1)^2}{2}\,d\theta = \int_0^{\theta_0} \frac{2n-1}{2}\,d\theta = \frac{2n-1}{2}\theta_0$$
The distance from the origin $r$ is within the interval $I = [n-1, n]$. The $r$ in the integral comes from computing the area integral in poolar coordinates.
