I have to verify a point:
I'm supposed to find the area of the region given in polar coordinates
$$\sec{\theta}\le r\le 2\cos{\theta}$$
I plotted the curves of $\sec{\theta}$ and $2\cos{\theta}$ between $-{\pi/4}$ and ${\pi/4}$
This gave me an indication that the region mentioned at the start comes from taking the area of $\sec{\theta}$ from $2\cos{\theta}$ but what I don't understand is how I can verify that $r$ is greater than or equal to $\sec{\theta}$, as was mentioned at the start.
Since $r$ is allowed to vary between $\sec\theta$ and $2\cos\theta$, the polar area would look like $$B' = \int_A \int_{\sec\theta}^{2\cos\theta} 1 \mathrm dr \mathrm d\theta$$ Where $A$ is the set $\{\theta \in (-\pi,\pi) : \sec\theta \le 2\cos\theta \}$ Now note that the transformation from carthesian coordinates to polar coordinates has determinant of $r$, so the true area is given by $$B = \int_A \int_{\sec\theta}^{2\cos\theta} r \mathrm dr \mathrm d\theta$$ I'm sure you can evaluate the inner integral; for the outer integral you might ask a new question once you found $A$.