The point t=0 is usually a particular point we like to label as time zero for convenience. It rarely has any significance other than providing a label for a typical starting time. If an event starts at t=-12 and ends at t=-2, it is no different than an event starting at t=0 and ending at t=10. Both things happen with time flowing in the positive direction and over the same time difference.
Sometimes people will talk about time flowing backwards and this would be moving in the negative direction, but again, the place you start is usually not very relevant. It simply provides a label or reference point. Essentially, time difference is typically the important thing to worry about.
This question seems related to the question you asked about integration and you thought that integrating on the negative side of the x-axis resulted in negative area. In both cases you should think more about the direction you are traveling (whether distance or time) and not on which side of the axis you are on. When you go to the right you are increasing in value and when you go to the left you are decreasing in value. For both time and distance, the difference of values between two points matters more than the actual values at those points.
Of course, I'm kind of mixing physics into the math here and your question is really as much of a math question, but I do this deliberately to help you visualize the right way to look at it.