This is an exercise from Apostol's Calculus, Volume 1. It asks us to sketch the graph in polar coordinates and find the area of the radial set for the function:
$$f(\theta) = \theta$$
On the interva $0 \leq \theta \leq 2 \pi$. I think to find the area we should just integrate $\theta \ d\theta$ from 0 to $2\pi$ like any other function? Is that right? Also I'm not sure how to think about sketching a function in polar coordinates.
The problem is the book gives the answer as $4\pi^3/3$ which is not what I get if I just integrate the function.