Question

I want proof for the relation between the area under the curve and the anti-derivative.
I don't understand how eventually the Riemann sum turned into the definite integral(anti-derivative) .

Answer 1

Check out this animation. It might give you an idea as to how it works.

Essentially, the Riemann sum is a finite sum of the area of \(n\) rectangles. As \(n\) increases, the area becomes more and more refined. Taking the limit as \(n\) approaches \(\infty\), the area becomes precise; this infinite sum is defined as the definite integral.