Concepts explained
What Integration Is — Slice Everything and Add It Up
If differentiation looks into an instant, integration does the opposite: it adds up countless small pieces to recover the whole. That these two operations undo each other is the heart of calculus.
There is a way. Slice the land into vertical strips.
Cut it into strips one metre wide and each strip is nearly a rectangle. The river end tilts a little, but over one metre the error is small. Measure each strip's height, work out its area, and add them all up.
Something odd happens at the limit. Make the strips infinitely narrow and each strip's area approaches zero, while the number of strips grows without bound. You are adding up zeros forever — and yet the total closes in on one exact value.
What has this to do with differentiation?
Differentiation asked how fast something is changing now. Integration asks the opposite: how much has accumulated.
Why that matters: slicing and adding is laborious, while differentiating is mechanical once you know the rules. If the two undo each other, the laborious problem can be swapped for the easy one — adding infinitely many pieces becomes differentiating backwards.
A little further in
A rectangular field's area is length times width. What about a bend of land along a river? There is no formula for an area bounded by an arbitrary curve.
But there is a method: slice it and approximate with rectangles. Pack narrow rectangles in and you get close to the true area — and the narrower the slices, the closer you get.
The method, two thousand years ago
Archimedes was already doing this. He found the area of a segment cut by a parabola: fit a triangle inside, fill the remaining gaps with smaller triangles, and those gaps with smaller ones again.
His answer was exact. But the method needed a fresh trick for every shape; what worked for a parabola did not carry over. It was an art to be reinvented each time, not a procedure anyone could follow.
Kepler used something similar. His problem was the area a planet sweeps out in a given time along an elliptical orbit — and he applied the same approach to gauging the volume of wine casks, slicing them into thin discs and adding.
They were one thing
The decisive step was not a new trick for areas. It was the discovery that finding areas and finding slopes undo each other.
- Think of the area under a curve, with the right-hand edge sliding.
- Push the edge a little and the area grows by a thin vertical strip.
- That strip's area is (height of the curve) × (how far you pushed).
- So the rate at which area grows is exactly the curve's height there.
- Differentiating the area gives back the original curve.
This is the fundamental theorem of calculus. To find an area you need not slice and sum: find a function whose derivative is the curve. A problem of adding infinitely many pieces became a problem of differentiating backwards.
Beyond area
Integration is not only about area. Anything with the structure of slicing finely and adding is an integral.
| Slice what | Sum gives |
|---|---|
| Speed at each instant | Distance travelled |
| Force at each instant | Work done |
| Thin discs | Volume of a solid |
| Density at each point | Total mass |
| Slices of probability | Chance of falling in a range |
Fourier went further. Any complicated waveform can be written as a sum of simple sine waves — and the way to find how much of each is present is an integral. Audio compression, image compression and telecommunications all stand on that calculation.
The question that remainsInfinitely many things were added and the total is finite. How did we come to accept that endless addition can have an end?