Answer
$0$
Work Step by Step
The parametric equations are:
$ x=2, y=\sqrt 5 \cos t, z= \sqrt 5 \sin t$
Here, we have
$dx=0 dt, dy=-\sqrt 5 \sin t dt, dz= \sqrt 5 \cos t dt$
Plug the above values in the given integral.
$\oint_C F \cdot dr=\int_{0}^{2 \pi} (12 \sqrt 5 \cos t) (0 dt) +(-\sqrt 5 \sin t dt) (9) +(45 \sin^2 t) (\sqrt 5 \cos t dt)$
$\int_{2 \pi}^{0} (-9 \sqrt 5 \sin t+45 \sqrt 5 \sin^2 t \cos t) dt= [9 \sqrt 5 \cos t+\dfrac{45 \sqrt 5}{3} \sin^3 t]_{0}{2 \pi}$
Thus, $(9 \sqrt 5 \cos (2 \pi)+\dfrac{45 \sqrt 5}{3} \sin^3 (2 \pi) )-(9 \sqrt 5 \cos 0+\dfrac{45 \sqrt 5}{3} \sin^3 0)=9\sqrt 5-9 \sqrt 5 =0$
