Visualization of the Fundamental Theorem of Calculus

Given a function f that is continuous on an interval [a,b], we can integrate the function from c to x such that a<c<b and a<x<b.

Now let F(x) on [a,b] represent the resultant general but particular anti-derivative of f. Now, F(x) represents the sum of area under the curve f from c to x. It’ll be positive if x>c and negative if x<c. For example, as you increase x, the ending point of the area extends to the right (positive x-axis). As you decrease it, it shrinks to the left (negative x-axis); at some point, it becomes negative and extends to the left negatively.

Furthermore, F(p)F(q) where a<p<b and a<q<b is a difference of area under the curve f starting from c to each p and q. In addition, consider dFdx. dF is a small area under the curve f whose height is f and whose width is dx. Thus, dF=fdx. Then dFdx=fdxdx=f as given.