# Visualization of the Fundamental Theorem of Calculus

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

Now let $F\left(x\right)$ on $\left[a,b\right]$ represent the resultant general but particular anti-derivative of f. Now, $F\left(x\right)$ represents the sum of area under the curve f from c to x. It’ll be positive if $x>c$ and negative if $x. 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, $F\left(p\right)-F\left(q\right)$ where $a and $a 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.