Discussion
The area swept by a polar curve \( r = f(\theta) \) between \( \theta = \alpha \) and \( \beta \) is
Answer: A
An element of the region is a thin sector of radius \( r \) and angle \( d\theta \), whose area is \( \tfrac{1}{2}r^{2}d\theta \), so
\[ A \;=\; \frac{1}{2}\int_{\alpha}^{\beta} r^{2}\,d\theta. \]
Setting \( r = a \) and integrating over a full turn returns \( \pi a^{2} \), which is the check worth doing to fix the half in memory
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