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 $A^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }A(x)}{\mathrm{\Delta }x}=f(x)$.