# 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 $$\frac{dF}{dx}$$. dF is a small area under the curve f whose height is f and whose width is dx. Thus, $$dF=fdx$$. Then $$\frac{dF}{dx}=f\frac{dx}{dx}=f$$ as given.