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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 _______________.

  1. $0$
  2. $1$
  3. $8 + 2 \pi$
  4. $ – 1$
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