Find the area enclose by one loop of the curve
r= 2 Cos[x] - Sec[x]
We need the limits of integration first. since the loop goes through the origin, we need to find the values of x for which the curve passes through the origin, so set r = 0
we have, $\displaystyle 2 \cos x - \frac {1}{ \cos x} = 0$
$\displaystyle \Rightarrow 2 \cos^2 x - 1 = 0$
$\displaystyle \Rightarrow \cos x = \pm \frac {1}{ \sqrt { 2 }}$
$\displaystyle \Rightarrow x = \frac {\pi}{4}, \frac {3 \pi}{4}, \frac { 5\pi}{4}, \frac {7 \pi}{4}$
looking at the graph, we see we want the region between $\displaystyle \frac {\pi}{4}$ and $\displaystyle \frac {7 \pi}{4}$
we want to go anticlockwise from $\displaystyle \frac {7 \pi}{4}$ to $\displaystyle \frac {\pi}{4}$ to get the desired area, but we must go from a smaller to a larger angle. so rewrite $\displaystyle \frac {7 \pi}{4}$ as $\displaystyle - \frac { \pi}{4}$ and we are in business
So our desired area is given by
$\displaystyle \int_{- \pi / 4}^{ \pi / 4}\frac {1}{2} r^2 ~dx$
$\displaystyle = \int_{- \pi / 4}^{ \pi / 4}\frac {1}{2} \left( 2 \cos x - \sec x \right)^2 ~dx$