Moving intervals

The title is very vague but I can't think of the right way to phrase it.
My issue is that I have an area under a graph and I would like to animate the upper end of the interval to move back and forth as to make the area move back and forth.
What I mean is how Grant did in Essence of Calculus, Chapter One,
https://youtu.be/WUvTyaaNkzM?t=634
like this.
How do I update the end of the interval to move the way he does?
I have no code to give, apologies, but I can't figure out how I would do it.

I tried figuring out how he did it in the old eoc project file, but I couldn't.