When evaluating integrals, if you convert to polar/cylindrical coordinates, I know you have to include $r$ ($r\, dr \, d\theta$).
However, when you parametrize first (for line integrals, or surface integrals) do you still include $r$? For example, I'll parametrize $x = r \cos\theta$, $y = r \sin\theta$, $z = z$. Plug in as $F(r(r,\theta))$, and do the cross product of the partial derivatives. Is the extra $r$ already included in this process?
In the conversion $dxdy=rdrd\theta$, the factor of $r$ is a Jacobian determinant. This generalises the result $du=u'dx$ in a single-integral substitution. You always need to include Jacobians; the real question is what the Jacobian should be for a particular problem. In general, a switch between two sets of variables $u_i,\,v_j$ has $d^n u = |\det J|d^n v$ with $J_{ij}:=\frac{\partial u_i}{\partial v_j}$. I recommend proving $dxdy=rdrd\theta$ as an exercise.