I was able to determine (c) is compact since $(e^{-x}\cos(x),e^{-x}\sin(x))$ spirals back to the x-axis and the horizontal segment bounds the area. But does the curve keep spiralling making it unbounded outside the horizontal segment?
For (d), I tried to plot the parametric surface in Mathematica, but all I got is a blank plot. Is there an algebraic way to do this? My best attempt was (let $x = u$ and $\theta = v$)
$x^2 + y^2 = e^{-2u}(\cos^2v + \sin^2v) = e^{-2u} \to 0$
EDIT: For (c), wouldn't it be not compact since it isn't the intersection, but the union?
EDIT2: Let me clarify. If it was the intersection of the two sets, I get only the point where they meet and that is only one point. If it were the union, I get the set of all points on the curve and the first set is clearly not closed (as demonstrated in (d) ). So I don't think the set is compact. Am I wrong?