Cavalieri's principle · the idea behind Hubbard 4.5 (Fubini)

Same cross-section at every height ⇒ same volume (or area). Lean it, shear it, slice it.

The idea, and how to show it to a student

If two solids have equal cross-sectional areas at every height, they have equal volumes. Think of a deck of cards. Push it into a slanted, wavy pile and the volume doesn't change, because every card is still the same card. If you have a deck or a stack of coins at hand, start with that.

A suggested order:

  1. 2D shear with a real deck of cards.
  2. Pyramids, to see that volume means adding up slices, and to get ∫₀¹ x² dx = ⅓ for free.
  3. Sphere, so a famous formula falls out.
  4. Then the napkin ring, Steinmetz solid and ellipsoid for fun.

After that, an integral is just a way to write down and calculate "add up the slices" when you can't find a matching solid to compare against. Turn on Stack of cards / coins on any 3D tab and you are looking at a Riemann sum.

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