Split radius R to get thinner rings, like using "graduated cookie cutter" to cut thin rings of dough. Split R forever, we'd have infinite rings, each infinitely thin. See how the estimated circle area improves with thinner rings. Each ring has width dr (difference in radius) and length 2πr so has area 2πr(dr). Rectangles show each ring's area. Sum of all rings' areas is the circle's area. Sum of all drs is R, even when dr shrinks to 0. As dr shrinks, total rectangle area better estimates πR².
Use smaller and smaller cookie cutters (left) to cut thin rings, each of thickness dr. If we could cut infinitely thin rings of dough and lay each out as a rectangle, we could use that to estimate the total amount of dough rings. When dr → 0 (dr goes to zero), the total rectangle area (rings laid flat) estimates total circle area (πR²) perfectly. This is calculus made simple. As dr → 0, the total rectangles area form a triangle whose area is half of 2πr X r. So that's πr². This is essentially calculus. A couple days ago, I showed calculus as thinning pie slices (triangles) as estimate of circle's area.
