I just wanted to have a visual intuition on how the Fundamental Theorem of Calculus works. Maybe it's not rigorous, but it could be helpful for someone (:.
I've used the concepts of velocity, distance and time because, I think, they are the ones with which we are more familiarized and also used them to write one part of the theorem (is it illegal?).
INSTRUCTIONS (::

Plug in any function you want in the input box v(t). Make sure that the variable you're using is t instead of x, though.

Choose your own interval by moving the points a and b along the x axis.

The general form of an antiderivative is a + C, where C is a constant. Move the slider ''Constant'' to shift the position of d(t) along the y axis. The slider has two purposes:

It allows you to move d(t) when it overlaps with v(t), so that the visualization doesn't get messy.

It shows you that changing the constant C doesn't change the difference d(b) - d(a): A neat way to see how the Definite Integral of v(t) is connected to its infinitely many antiderivatives (Remember that the antiderivatives of a function just differ in C, the constant term).

Check the box ''Riemann Sum'' and move the slider ''Rectangles'' to see how it approaches a Definite Integral as the number of rectangles goes to infinity (the slider doesn't go to infinity but up to 60 isn't that bad).