Choosing a Gaussian surface is not very hard, but it is fairly subtle. Unlike the complicated surface shown in the introduction, we are going to choose simple surfaces for which it is easy to calculate the flux. There are two parts to the flux integral. First, the magnitude of the field can vary. To simplify this aspect, we want to choose a surface for which the field has constant magnitude.
Next, the dot product can vary if the angle between the normal to the surface and field vector changes for different area elements dA. Remember the dot product gives EdA cos(theta). We want to choose a surface for which the angle remains constant. The simplest angles are those for which E and dA are parallel and those for which E and dA are perpendicular. We will often use these.
Next go to Gauss' Sphere1.