here is my suggestion for one of the most basic ideas of calculus: if two plane figures are such that all horizontal slices have the same length, then they have the same area.

it follows that two triangles of the same base and height have the same area.

the next case of this principle is that two solids all of whose horizontal plane slices have the same area, have the same volume. archimedes used this cleverly to show that a sphere inscribed in a cylinder has 2/3 the volume of that cylinder.

in a calculus class they combine this with algebra to show how to compute a formula for the volume formula of a figure from the formula for the slice area.

by the way, a mathematician is not someone who knows a lot of math facts, but someone who knows how to deduce them. I.e. the prime compliment for a mathematician is that she "knows nothing but can do anything".

Last edited by mathwonk; 08/21/12 06:28 PM.