Chapter 14 - Vector Calculus - 14.3 Conservative Vector Fields - 14.3 Exercises - Page 1085: 34

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We are given that $F(x,y)=(y, \ x)$ Also, the curve is $r(t)= (8 \cos t, 8 \sin t)$ This implies that $r'(t) = (-8 \sin t, 8 \cos t)$ Therefore, the integral is: $\oint F \ dr=\int_0^{2 \pi} F[r(t)] r'(t) \ dt\\=\int_0^{2 \pi} (8 \sin t, 8 \cos t) (-8 \sin t, 8 \cos t)\\=\int_0^{2 \pi} 64 \cos (2t) \ dt \\=32 \times [\sin (2t)]_0^{2 \pi}\\=0 $