Why is the area of ​​a curvilinear trapezoid the difference of antiderivatives?

Question

Mathematics

Hundreds of explanations of what a definite integral is are shown by columns, of which the sum of infinitesimals and suddenly - they draw the sign of a definite integral and say that it can be calculated by the difference of antiderivatives. But why? How to get there? Why the primitives? Well, and, accordingly, why is the area an integral, which is calculated in this way? Just don't answer "by definition"...

Answer 1

What is an integral? If we recall the school explanation, then this is the "stupidly" sum of all the rectangles that are placed inside the square, which are drawn in each textbook. The notation says "the sum of each value of the function at each point in infinitesimal increments", so this dx literally means infinitesimal x. It kind of says mathematically "shift the next value we find by a little bit". In essence, we find the sum of all the line heights that can be crammed under the function.
Why exactly Newton Leybinz's formula works, I advise you to look at the proof, to sort it out on your fingers. Understand what the sum limit means.
I tried to explain on my fingers, the mathematical explanation may be incorrect, so be sure to look for proofs of all formulas,

Answer 2

G. M. Fikhtengolts. Fundamentals of mathematical analysis. T. 1. M.: Nauka, 1968. P. 156.

Answer 3

First you need to understand what a derivative is - in fact, it is the rate of change of a function. And the antiderivative - just the opposite - how the value is accumulated if the speed is given.
For example, there is a pool into which water flows at a speed v1(t) = 5. Then the water level in the pool h1(t) = h0 + 5*t (<< and this function is the antiderivative)
Or if the speed v2(t) = t , then h2(t) = t*t/2
You need to build a table with your hands and draw a graph - everything will become clearer.