In the early days of our learning , most of our knowledge and practice of Integration is based on the Fundamental Theorem of Integral Calculus (FTIC). But the ideas and the intuition of it changes drastically as one tries to move onto higher dimensions or onto more "general sets", leading to new intuition, new Mathematics, entangling analysis with topology. In even more abstract setting, a rephrasing of FTIC leads to new theories of Integration with its "associated FTIC" .
Professor Kalyan Bidhan Sinha
The Fundamental Theorem of Integral Calculus