The area of the loop of strophoid $r = a(\sec\theta - \tan\theta)$

Question

Find the area of the loop of strophoid $$r = a(\sec\theta - \tan\theta)$$.

I tried finding the limits of integration for polar coordinates but once I equate it to zero, it returns $\frac{\pi}{2}$. As well as taking its symmetric $r = a(-\sec\theta - \tan\theta)$ then shifting it by $a$ with respect to the polar axis, it returns $\frac{\pi}{2}$. We know that both $\tan\theta$ and $\sec\theta$ diverges at $\frac{\pi}{2}$. I appreciate some help.

Answer 1

Note the convergence as $\theta\to \frac\pi2$,

$$\sec\theta - \tan\theta = \frac{1-\sin\theta}{\cos\theta}= \frac{\cos\theta}{1+\sin\theta}\to 0 $$