Recently, I am reading the book Calculus by Michael Spivak as usual and made it to the Fundamental Theorem of Calculus. While I understand the steps of the proof, I found it hard to develop the intuition and the meaning of the Theorem which states as follows:

If we investigate the proof of the first part, assuming some knowledge of the definition of differentiation using limit, and the definition of integral as the lower sum and upper sum of any partition of [a,b], it is not hard to follow the proof. But I feel like I do not appreciate enough the “fundamental” part of the theorem. What makes it so fundamental? Then I came across a very good paper titled: Historical Reflections on Teaching the Fundamental Theorem of Integral Calculus by David M. Bressound. The article explores how mathematicians discovered this theorem, the intuition behind it, and how the theorem is being refined to be expressed in the above form.

In this blog post, I will present another piece of proof not covered in the article above from Barrow which is very similar to the Leibniz proof mentioned. The reference of the proof can be found in the article titled: Barrow And Leibniz on the Fundamental Theorem of The Calculus.

The proof made use of the following diagram to show that DT = R * DF/DE, which relates the tangent line (DT: rate of change), area (DF: integral), and the function itself (DE). In fact, DT is called the subtangent and FT is the tangent line. But they are closely related by the triangle TDF. I will walk you through the graphical proof step by step.

Of course, this is just a graphical proof, and Barrow further gives rigorous proof. The appreciation of this graphical proof is that the rate of change (Tangent) of the area function (A) and the function (f) itself are related.

In this next blog post, I want to cover the Issac Newton intuition on the theorem.