Area Enclosed by First Turn of Archimedean Spiral

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

The area $\mathcal A$ enclosed by the first turn of $S$ and the polar axis is given by:
 * $\mathcal A = \dfrac {4 \pi^3 a^2} 3$


 * ArchimedeanSpiralArea.png