I found this in Schaum's outline of Advanced Calculus stating: Cases
in case (b),
For a closed curve PQRSP not surrounding the origin, $\phi=\phi_{0}$ at $P$ and $\phi=\phi_{0}$ after a complete circuit back to $P$. In this case, the line integral equals $\int_{\phi_{0}}^{\phi_{0}} d \phi=0$.
How is it possible? when
in case (a),
For a closed curve ABCDA surrounding the origin, $\phi=0$ at $A$ and $\phi=2\pi$ after a complete circuit back to $A$. In this case, the line integral equals $\int_{0}^{2\pi} d \phi=2\pi$.