Path Integral equals zero on non conservative field

Question

I was doing some excercises and I was asked to compute the line integral along certain path. I used greens formula to calculate the work. When computing the integral I had to divide the domain in two sections and one of those domains equals zero. Is this possible?

Here is what I have done so far (its question 1.2)

]Question:

Answer 1

For a conservative field $\vec{F}$ the curve integral $\oint_\gamma \vec{F} \cdot d\vec{x} = 0$ for every closed path $\gamma$.

If a field is not conservative, the integral can still be zero for a lot of closed curves, but not for every curve.