Answer
$0$
Work Step by Step
We are given that $F(x,y)=(x, \ y)$
Next, the integral can be written for the given curve into three parts as:
$\oint F \ dr=\int_0^{1} [F[r(t)] r_1'(t) +F[r(t)] r_2'(t)+F[r(t)] r_3'(t)]\ dt\\=\int_0^{1} [(0, 1-2t) \times (0, -2)+(t,t-1) \times (1,1) + (1-t, t) \times (-1, 1)]\ dt \\=\int_0^{1} (8t-4) \ dt \\=4 \times [t^2-t]_0^{1}\\=0 $
