space-time is not "nothing" ... space-time is a stage on which matter, energy and time play out... Why it has negative pressure and 'expands' is still unknown: but we can say we observe that this is a characteristic of space-time.
I think that's about as far as physics will take you.
If you consider the "line" to be space-time itself, you have the idea.
You can easily extend this idea to 2D by considering the surface of a balloon. Blow it up a little, use a marker to create dots on the surface... Blow it up more: you will notice that every dot moves away from every other...
Space(-time) is a "thing" ... it has dimensions and (from our measurements of the expansion) a negative pressure. The more space-time you have, the more negative pressure. Hence the 'expansion' of space accelerates. This is what we are measuring.
We don't yet have a good handle on the source...
Oops! My bad: symmetric is the wrong word. As you realized, Sin reflects THRU the origin -- about both x and y... not just the y-axis as I implied.
Sorry!
Since Cos[t] = Cos[-t], the integral should be identically 0. If you think about it, Sin is symmetric about the origin: hence any definite integral centered on 0 (i.e. - {-t -> t}) MUST 'sum' to 0.