Originally Posted by mathwonk
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".

Not to rain on your parade, but the basic of idea of calculus is to sum up an infinitely small number of things to get a finite quantity, and then to do this in reverse, divide a finite quantity into an infinite number of small things. Once you can show that you can do this reliably, then the next trick is to extend this idea to where you can sum up across functions that produce an infinite number of small things. Historically, the basis of this approach had its root in addressing Zeno's Paradox.

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

As for your statement that a mathematician does not know a lot of facts. Its the exact opposite. Mathematicians know an enormous amount of facts and know how to operate a lot of intellectual tools, too. Its this combination of knowledge and operative ability that allows them to work a large set of problems and to attack unknown ones.