Area between Radii and Whorls of Archimedean Spiral

Theorem
Let $S$ be the Archimedean spiral defined by the equation:
 * $r = a \theta$

Let $\theta = \theta_1$ and $\theta = \theta_2$ be the two rays from the pole at angles $\theta_1$ and $\theta_b$ to the polar axis respectively.

Let $R$ be the figure enclosed by:
 * $\theta_1$ and $\theta_2$
 * the $n$th turn of $S$ and the $n+1$th turn of $S$

The area $\mathcal A$ of $R$ is given by:
 * $\mathcal A = ...$

Proof
The straight line boundaries of $R$ are given as $\theta_1$ and $\theta_2$.

The corners of $R$ are located where:
 * $\theta = \theta_1 + 2 n \pi$
 * $\theta = \theta_2 + 2 n \pi$
 * $\theta = \theta_1 + 2 \left({n + 1}\right) \pi$
 * $\theta = \theta_2 + 2 \left({n + 1}\right) \pi$