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.
