User:IssaRice/Fundamental theorem of calculus

There's a typical picture of FTC1 that you see in places like Pugh's analysis book or 3Blue1Brown's video on FTC. This explanation makes sense, but I want point out a few different ways of thinking about the picture. One is, like the 3b1b video says, to look at the incremental area. You get $\Delta A(x)\approx f(x)\Delta x$. So then you divide by $\Delta x$ and take the limit as $\Delta x\to 0$ and get $A'(x)=\lim _{\Delta x\to 0}{\frac {\Delta A(x)}{\Delta x}}=f(x)$.

Another way of looking at this that I saw in one of John Stillwell's books is that $\int f(x)\,dx$ is the sum of infinitely many quantities $f(x)\,dx$. So the incremental thing you add is $f(x)\,dx$, i.e. $d\int f(x)\,dx=f(x)\,dx$. Now if you divide by $dx$ you get ${\frac {d\int f(x)\,dx}{dx}}=f(x)$, which again is FTC1.

Finally, let's look at $A(x)=\int _{a}^{x}f(t)\,dt$. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, as the width of the rectangle changes, is the height of that rectangle".

Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they don't even mention how to visualize FTC2. Like, they probably don't have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be visualized using the same picture as FTC1, but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!