I'm not understanding something well in 3Blue1Brown's Essence of Calculus Chapter 1 video.
At 5:23, when we drew a graph, how did we figure out that we had to draw the graph of the function $2\pi r$? Was it just a guess? I feel like I'm missing something important.
The graph comes from the formula of the circumference, $2\pi r$. What he's demonstrating is unraveling each infinitesimally small circumference and graphing them, to create a triangle: the area under the graph of $2\pi r$.
Thinking about this non-rigorously, you can think of the area of a circle as a collection of infinitely many concentric circumferences, each infinitely thin (and thus the circumference of infinite circles). Then, taking these circumferences, we can show that the sum of these infinite parts is the area under the graph $2 \pi r$, the circumference formula and also the area of the original circle. This is the basic idea of integration.
Thus, integrating $2 \pi r$, or finding the area under the graph of the circumference, leads to the area of a circle, or $\pi r^2$. We can also observe that $2\pi r$ is the derivative of $\pi r^2$. This connects integration and differentiation down the road.