What is the proof of the definition of definite integrals? Why is there subtraction of antiderivatives in the first place?

Question

What is the proof of the subtraction of the Antiderivatives?

Answer 1

It's not so much a proof as it is a definition.

Given a function f(x) that is continuous on the interval [a, b], you can divide the interval into n subintervals of equal width, each Δx wide, and for each interval choose a point, x_i where x₁ = a, x₂ = a + Δx, x₃ = a + 2Δx, . . . and x_n = b, then:

So pictorially, f(x_i)•Δx is the area of the rectangle that is f(x_i) high and Δx wide and you are adding an infinite number of these rectangles from x = a to x = b. The result is that you have calculated the area under the curve of f(x) from x = a to x = b