= Solution
{c}
In <cylindrical coordinates>, $(-y,x,0)=r\mathbf e_\theta$ and $x+r=r(1+\cos\theta)$. Hence the <vector field> simplifies to
$$
\mathbf F=\frac{\sqrt{1+\cos\theta}}{r}\mathbf e_\theta.
$$
Take the <orientation> of $C$ to be increasing $\theta$, as in its given <parametrization>. On the <circle>, $d\mathbf r=a\mathbf e_\theta\,d\theta$, so its <line integral> is
$$
\int_C\mathbf F\cdot d\mathbf r
=\int_{-\pi}^{\pi}\sqrt{1+\cos\theta}\,d\theta
=\sqrt2\int_{-\pi}^{\pi}\cos(\theta/2)\,d\theta
=\boxed{4\sqrt2}.
$$
We used $\sqrt{1+\cos\theta}=\sqrt2|\cos(\theta/2)|$; the <cosine> is nonnegative on this interval. In particular, the answer does not depend on $a$.
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