Parametrization of a cylinder that is parallel to x axis The answer is no it does not matter.
The surface is $y^2+z^2=4$, I parametrized it so:
$\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$
But Pauls Outline works through the problem with the j and k components switched. Does it matter which way I go about parametrizing this surface?

It really doesn't matter. Thinking of a similar problem in two dimensions, both
$$\cos\theta\,\mathbf{i} + \sin\theta\,\mathbf{j}\text{ and }
  \sin\theta\,\mathbf{i} + \cos\theta\,\mathbf{j}$$
parametrize the unit circle, but in different ways.
