Dynamic Interval Merger

You are tasked with designing and implementing a data structure that efficiently manages a dynamic set of integer intervals and computes the total coverage length (i.e., the length of the union of all intervals).

Problem Description:

• Each interval is represented as a pair [start, end] (inclusive). Intervals may overlap.

• Implement the following operations:

  • addInterval(interval): Adds an interval to the data structure.
  • getTotalCoverage(): Returns the total length of the union of all added intervals.

For example, if the following intervals are added in order:

1. addInterval([1, 5])
2. addInterval([3, 7])
3. addInterval([10, 12])

Then, getTotalCoverage() should return 9 (since the union covers from 1 to 7 and from 10 to 12, with overlapping segments counted only once).

Requirements:

• Design the data structure to be efficient in terms of both time and space, considering that a large number of intervals may be added.
• Ensure that the add and query operations can handle dynamic updates.
• Write clean, maintainable code and consider edge cases (e.g., intervals that fully overlap each other, intervals that touch at the boundaries, etc.).

Implement your solution in the programming language of your choice.