GATE EC 2022 Set 1 Q1 — Vector Analysis

Consider the two-dimensional vector field $\overrightarrow{\rm F}(x, y) = x \overrightarrow{i} + y \overrightarrow{j},$ where $\overrightarrow{i}$ and $\overrightarrow{j}$ denote the unit vectors along the $x –$axis and the $y –$axis, respectively. A contour $\text{C}$ in the $x – y$ plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral

$$\oint_{c}\overrightarrow{\rm F}(x, y) \cdot (dx \overrightarrow{i} + dy \overrightarrow{j})$$

is _______________.