Understanding the fundamental theorem of calculus via squeeze theorem I was just taught "integration is the opposite of differentation, this will give the area"
While I can integrate, I am not sure why it works and the "anti derivate" magically is the area. I really need some help getting this idea in my head. I am using https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus for reference.
If we want A(x), I am not sure how adding an extra red section helps. I think I understand the red area is defined as approximateley A(x)-A(x-h), which equals f(x)h.
Then I have read, this is between
hf(x)< RED AREA < f(x+h).
I then see we divide by h, and are left with the defintion of differentation when h>>0.
However I have 2 questions'
1) Surely we are not concerned with the red area, we want the blue area
2) When we take the limit as h>>0 this leaves an infitesimal slice (I understand that integration is the summation of an infinate amount of infiniately small sliced) but the concept of adding +h to the area (thus creating a red area) only to remove it doesn't seem to have any effect on the blue area.
Thanks a lot for your help. Hopefully its something easy I missed, and I can get there!
 A: I have the feeling that a part of your confusion comes from the fact that you're reading a paragraph of the Wikipedia-page with an expectation that's different from its goal. They don't define the integral in that paragraph.

2) When we take the limit as h>>0 this leaves an infitesimal slice (I understand that integration is the summation of an infinate amount of infiniately small sliced) but the concept of adding +h to the area (thus creating a red area) only to remove it doesn't seem to have any effect on the blue area.

You are right: you don't need the red part to understand what the integral does, where it comes from, how it is defined. As you say: you divide the interval into subintervals and you build rectangles whose length you let go to 0. Adding the areas of the rectangles gives an approximation of the area under the curve and we get (or: define!) the real area by taking the limit of $h \to 0$ (I'm omitting details here). This is done in many books, websites and partly also in the Wikipedia article.
However, you're referring to the picture and the reasoning in the paragraph Geometrical meaning and there, they are trying to explain what the relation is between actually calculating this area and the concept of derivatives and anti-derivatives. More precisely: it gives you the relation between $f$ and the area function $A$, and they're connected as one being the derivative of the other one.

Thanks, I can see that if I differentiate A(x), I will get f(x), but may I ask why you want to differentiate A to get the area, when A is the area? What is the logic or intuition in this?

So what they achieved in that paragraph is showing that you can obtain a formula for the area under the graph of $f$, if you can find a function $A$ that satisfies $A'(x) = f(x)$, what we call an anti-derivative (or primitive function) of $f$. Note that you don't necessarily "discover" this relation if you only define the integral as the limit of a sum of areas of rectangles.
