electromagnetism - Ampere's circuital law and the type of surface to be used

What kind of a surface can we use for Ampere's circuital law? I was taught that any enclosed surface can be used for Gauss's Law(something like a cube,a sphere)-essentially 3-D enclosed surfaces.

For Ampere's law, I have used a circular ring for calculation of Magnetic field due to wire(and also rectangular shapes for solenoids)-essentially 2-D enclosed surfaces like a ring,rectangle etc.

Now when I came to know about Maxwell's Displacement Current, the book uses a bucket kind of a shape like in the figure.

I am confused.One side is open and the other is closed.(I apologize if this sounds dumb.I only have a basic idea about the law.I don't really understand the 'surface integral' part and how it's supposed to be used.)

I've seen this link.I don't understand what is said there.

Answer

Neither S1 nor S2 is meant to be a closed surface.

S1 is just a disk, the shape of a dinner plate.

S2 is a truncated paraboloid. A bowl shape. The blue area in the drawing is not part of S2. Taken away from the rest of the drawing, it would look something like this:

(Image source: Wikemedia user Krishnavedala)

The whole point of this argument for displacement current is that it doesn't matter what shape the surface is, as long is its boundary (P1) is kept the same. You can take your initial flat disk that passes through the wire, and stretch it out so that it goes between the capacitor plates instead, and you should still obtain the same surface integral. Therefore there must be some kind of current passing between the plates, and we call this current displacement current.