$\backslash$end$\{$array$\}]$

Compare that computation with the line integral on the boundary of $.$ From the picture, notice that boundary consists of $4$ curves. Parametrize each curve by restricting the domain of $$ to an appropriate subset.

$C_1$ Straight line with $=0$

${}_1={\displaystyle \underset{G_1}{\overset{\phantom{}}{}}}F*dr{\displaystyle \underset{0}{\overset{1}{}}}0dr$

$=0$

$C_2$ Straight line with $=\frac{}{2}$

${}_2={\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{0}{\overset{1}{}}}dr$

$=$

$C_3$ Straight line with $r=0$

${}_3={\displaystyle \underset{C_3}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{0}{\overset{\phantom{}}{}}}d$

$=0$

$C_4$ Arc with $r=1$

$d$

${}_1={\displaystyle \underset{G_4}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{0}{\overset{\phantom{}}{}}}d$

$$