Adding Up Thin Slices
Last timeOne Rule Inside Another
A total is built by cutting the thing into slices thin enough that each one can be treated as constant, adding them, and asking what the sums settle on as the slices thin.
A car travels for an hour at a steady sixty miles an hour and covers sixty miles. One
multiplication. Now let the speed vary. The multiplication is no longer available,
because there is no single speed to multiply by, and no amount of cleverness will
produce one.
The way out is to stop asking for a single speed. Cut the hour into minutes. Within one
minute the speed barely changes, so one multiplication per minute is nearly right, and
sixty nearly right numbers added together is nearly the answer. Cut it into seconds and
it is nearer still. This lesson is about what that process converges to and why it
converges at all.
- the value of the rule somewhere on the i-th slice. Which point is chosen inside the slice stops mattering as the slices thin, which is itself something that has to be shown rather than assumed
- the width of a slice. Thin slices make each multiplication more nearly right and there are more of them to add, and the two effects do not cancel
- how many slices. Doubling it halves the width, and for an ordinary rule halves the error
- the total: a distance if the rule is a speed, an area if the rule is a height, a charge if the rule is a current. The construction does not care which
Trapping the answer
The estimate above is wrong by an unknown amount, which is unsatisfying. There is a
better move. On each slice, use the lowest value the rule takes there, and the sum is
certainly too small. Use the highest, and it is certainly too big. The truth is
trapped.
Here is that done for the squaring rule between zero and one.
| step | number of slices | sum using the lowest value | sum using the highest value | gap between them | what happened |
|---|---|---|---|---|---|
| 1 | 2 | 0.125 | 0.625 | 0.5 | Two slices is a crude start, and the answer is already known to lie between these. |
| 2 | 4 | 0.2188 | 0.4688 | 0.25 | Doubling the slices halves the gap, exactly. |
| 3 | 8 | 0.2734 | 0.3984 | 0.125 | And again. The two sums are closing on each other from above and below. |
| 4 | 16 | 0.3027 | 0.3652 | 0.0625 | A third of one is about zero point three three three, and it sits inside every row here. |
| 5 | 32 | 0.3179 | 0.3491 | 0.0312 | The gap is exactly one over the number of slices, every time. |
The squeeze is what makes the limit exist. Any number that is going to deserve the
name of the total has to be at least every lower sum and at most every upper sum. If
the gap closes, exactly one number qualifies. If the gap does not close, more than one
number qualifies and the total is not defined, which is not a flaw in the method but a
true statement about such a rule.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents