We ought never to allow ourselves to be persuaded of the truth of anything unless on the evidence of our own reason – René Descartes
Visualization of the Fundamental Theorem of Calculus
Given a function f that is continuous on an interval ,
we can integrate the function from c to x such that and .
Now let on represent the resultant general but particular anti-derivative of f.
Now, represents the sum of area under the curve f from c to x.
It'll be positive if and negative if .
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, where and is a difference of area
under the curve f starting from c to each p and q.
In addition, consider .
dF is a small area under the curve f whose height is f and whose width is dx.
Thus, . Then as given.