Matpic
mathematics, in pictures
Ladder

The integral — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

    Adding up rectangles until they stop being rectangles

    The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

    rung 1 · analysis
  2. Area is the undoing of slope. Above, a positive function with the area from 0 to 1.80 shaded. Below, that area plotted against where it stops. The lower curve's slope at 1.80 is 1.129, which is exactly the upper curve's height there.

    Area is the undoing of slope

    Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

    rung 2 · analysis

All ladders