Matpic
mathematics, in pictures
Series

The integral — the series

4 essays on 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.

    part 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.

    part 2 · analysis
  3. Area out to infinity, for three powers. Left: the curves 1/√x, 1/x, 1/x² from x = 0 to 10, with the region beyond x = 1 shaded under the lowest. Right: the area from 1 to T for each, on logarithmic scales, for T up to 10^6. 1/√x keeps growing, 1/x keeps growing, 1/x² levels off at 1.

    An endless region with a finite area

    A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.

    part 3 · analysis
  4. Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

    A rectangle cut by a curve

    Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

    part 4 · analysis

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