Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
Build volume by accumulating cross-sections
Problem
A pyramid is approximated by \(n\) layers whose \(k\)th slice area is \(B(k/n)^{2}\) and thickness is \(h/n\). Form the volume sum, use the supplied sum-of-squares factor, and take the limiting factor to derive the volume formula.
A slice area alone is not volume; each layer must also carry its thickness. We’ll form one thin-layer contribution, sum those contributions across the full height, use the supplied square-sum identity to simplify the position factors, and refine the layers until the remaining coefficient stabilizes.
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