Answer
$4\pi$
Work Step by Step
Here, $F(r(t)) \cdot \dfrac{dr}{dt}=\lt \cos t, 2 \sin t \gt \cdot \lt -\sin t,2 \cos t ,0 \gt =4 \cos ^2 t-\sin t \cos^2 t$
Now, $\int _C F(r(t)) \cdot dr =\int _0^{2\pi} 4 \cos ^2 t-\sin t \cos^2 t dt$
or, $=(\sin 4\pi-\sin 0) +2 (2 \pi-0)+\dfrac{1}{3} [\cos ^3 (2 \pi)-\cos^3 (0) ]$
or, $=4\pi$
