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 $\mathrm{\Delta }A(x)\approx f(x)\mathrm{\Delta }x$. So then you divide by $\mathrm{\Delta }x$ and take the limit as $\mathrm{\Delta }x\to 0$ and get $A^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }A(x)}{\mathrm{\Delta }x}=f(x)$.

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 $A(x)=\underset{a}{\overset{x}{\int }}f(t)dt$. 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, is the height of that rectangle".

Now, on to FTC2.