Find the area inside the polar curve $r^2=\cos(2\theta)$ I feel a bit silly asking this question as it is no doubt relatively simple, but it has been bugging me.
Given the polar curve described by $r^2 = cos(2\theta)$, find the area inside the curve. 
My intuition is to integrate from $0$ to $2 \pi$, but this clearly just yields 0. I have found that the area is 1, but this is just from dividing the curve into 4 pices and integrating from $0$ to $\pi/4$, then multiplying the resultant are by 4.
Why is it that the usual method of integrating $\frac{1}{2}\int_{0}^{2 \pi} r(\theta)^2 d\theta$ fails here?

The problem is the curve is not orientable. So when you integrate to a singularity point, wrong orientation makes the area negative. Hence the total area zero by symmetry. A similar example is integration of $f(x)=x$ on $[-\pi,\pi]$ leads to zero.
