Wow, I might finally grok this. At h=0, the area of the dome's slice = the cylinder's slice (πRR). As h increases, the dome slice gets smaller, but by exactly

Wow, I might finally grok this. At h=0, the area of the dome's slice = the cylinder's slice (πRR). As h increases, the dome slice gets smaller, but by exactly πhh. I picture that smaller circle in the plane of my screen, perp to the dome slice. Somehow, for any given h, that πhh makes up for the lost area at the edge. The bit about the Hat Box theorem still just makes me blink.