GATE Electronics and Communications (EC) 2022 Solved Paper

Consider the two-dimensional vector field  $\overrightarrow{\mathit{F}}\left(\mathit{x},\mathit{y}\right)=\mathit{x}\overrightarrow{\mathit{i}}+\mathit{y}\overrightarrow{\mathit{j}}$ , where  $\overrightarrow{\mathit{i}}$  and  $\overrightarrow{\mathit{j}}$  denote the unit vectors along the x-axis and the y-axis, respectively. A contour 𝐶 in the x - y plane, as shown inthe 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
${\oint }_{\mathit{C}}\overrightarrow{\mathit{F}}\left(\mathit{x},\mathit{y}\right).\left(\mathit{d}\mathit{x}\overrightarrow{\mathit{i}}+\mathit{d}\mathit{y}\overrightarrow{\mathit{j}}\right)$
is ________.