Why does the Fundamental Theorem of Calculus work?

Answer 1

Check out the video above.

And here's the link to the spot in the 3 blue 1 brown video I mentioned that goes really in depth on this: https://youtu.be/rfG8ce4nNh0?si=ZeSO_nfZyQxuh_q4&t=622

I highly recommend checking it out!

Answer 2

The FTC works because, at heart, integration is just a limit of sums of the form “height × width,” and differentiation measures how an accumulated sum changes when you tweak its endpoint. Continuity ties these limits together for Riemann integrable functions.

Answer 3

Use this link to look at the drawing I made to answer this question:

https://imgur.com/FhyFr2P

In the above picture, I created a function f whose graph consists entirely of line segments. Then, its derivative f' is a bunch of horizontal line segments, the heights of which are the slopes of the corresponding line segments. I considered the interval [0, 9]. Notice how computing the integral of f' on the interval [0, 9] required multiplying the height of each horizontal line segment of its graph in the interval by its length. So, for example, the second horizontal line segment in the interval had a height of 3, started at x=2, and ended at x=5. This means it has a length of 5-2=3, so the area it produces "under the curve" is 3*3=9.

Amazingly, this is same calculation that would be done to compute the change in height of f over the interval [2,5], given that its slope is 3. So, the sum of the signed areas under the graph of f' here can be seen to give

(f(2) - f(0)) + (f(5) - f(2)) + (f(6) - f(5)) + (f(9) - f(6)) = f(9) - f(0),

exactly what is predicted by the fundamental theorem of calculus. It might not be so surprising that this idea generalizes to functions like f(x)=x^3, so that finding the area under the curve in the graph of f'(x)=3x^2 on the interval [a,b] is b^3-a^3, since we could imagine approximating the function with line segments like this. In fact, the Mean Value Theorem is used in the proof of the Fundamental Theorem of Calculus, being the way in which a differentiable function can be treated like line segments!