Answer
$30 \pi$
Work Step by Step
We have: $x =5 \cos t-\cos 5t; dx=[-5 \sin t +5 \sin (5t) ] dt$ and $y=5 \sin t -\sin 5t; dy=[5\cos t -5 \cos 5t] dt$
Suppose that $F=(\dfrac{-y}{2}, \dfrac{x}{2})$
When $C$ is in the counterclockwise direction, we use: $A=\oint_{C} x dy=-\oint_{C} y dx$
and when $C$ is in the clockwise direction, we use: $A=-\oint_{C} x dy=\oint_{C} dx$
Now, $A=\oint_{C} F dr=\oint_{C} \dfrac{-y}{2} dx+\dfrac{x}{2} dy$
or, $=(1/2) \int_{0}^{2 \pi} [-(5 \sin t -\sin 5t)( [-5 \sin t +5 \sin (5t) ])] + (5 \cos t-\cos 5t) ([-5 \sin t +5 \sin (5t) ] dt) $
or, $=\dfrac{1}{2} \int_{0}^{2 \pi} 30-30 \sin t \sin 5t -30 \cos t \cos (5t) $
or, $=\dfrac{1}{2} \int_{0}^{2 \pi} 30 dt + \int_{0}^{2 \pi} -15[\cos (-4t) -\cos 6t] -15 [\cos (-4t) +\cos 6t) dt$
or, $= 30 \pi+[\dfrac{-15 \sin 6t}{4}]_0^{2 \pi}$
or, $=30 \pi$
