The Two Questions Are One Question

Last timeAdding Up Thin Slices

Accumulating a rate gives back the quantity it came from. That single fact turns a limit of sums into a subtraction, and it is why rates and totals are studied together.

Two lessons ago a rate was computed: the rate of the squaring rule is the doubling

rule. One lesson ago a total was computed: the total of the doubling rule up to an edge

is the squaring rule. Those were separate pieces of work using separate methods, and

they came out as exact mirror images of each other.

That is not luck. It happens every time, and this lesson is about why, because once it

is believed, almost every total anybody ever needs becomes a subtraction.

FIG 1The two halves of the connection

: the running total: how much has accumulated from the fixed start a up to the movable position x. It is a rule in its own right, built from f
: the rule being accumulated, at the position the running total has reached. The claim is that this is the rate at which the running total is growing
: any rule whose rate is f. There are infinitely many, all differing from each other by a constant, and the theorem does not care which one is used
: the subtraction that replaces the limit. Any constant added to F appears in both terms and cancels, which is why any choice of F works

The first statement says that accumulating and then taking the rate gets you back where you started. The second says that if you can find a rule whose rate is the one you want to total, the whole limit of sums collapses to a subtraction at the two ends. The second follows from the first in about three lines, and the first follows from one picture of a thin sliver. Both require the rule being totalled to be continuous, which every rule in this course is.

The one-line reason

Let the running total be how much has accumulated from a fixed start up to a movable

position. Now push the position forward by a small width. What does the accumulation

gain? One thin sliver. Its width is the small width, and its height is whatever the

rule is doing there, which across a thin enough sliver is nearly a single value.

So the gain is about the current value of the rule times the width. Divide the gain by

the width, exactly as the definition of a rate requires, and the width cancels, leaving

the current value of the rule. Let the width go and the word nearly becomes exact,

because a continuous rule varies by less and less across a vanishing sliver.

The rate of the running total is the rule. That is the whole argument, and it is worth

noticing how much it resembles the first lesson on rates: write the change in full,

cancel the width, then let the width go.

FIG 2Checking the sliver argument with arithmetic

step	position	total accumulated so far	gain over the next hundredth	gain per unit width	the rule here	what happened
1	0.5	0.25	0.0101	1.01	1	The gain divided by the width is one and one hundredth; the rule here is one.
2	1	1	0.0201	2.01	2	Two and one hundredth against two. The same overshoot of one hundredth.
3	1.5	2.25	0.0301	3.01	3	Three and one hundredth against three. The pattern is the width itself.
4	2	4	0.0401	4.01	4	Four and one hundredth against four, and shrinking the width would shrink the gap.

4 steps

The rule being accumulated is the doubling rule, so the running total from zero is the squaring rule. The last two columns are the test: the growth rate of the accumulation against the height of the rule at the same place. They differ by exactly the width used, one hundredth, every time, which is the roughness the argument promised would vanish. Use a thousandth instead and the fourth column reads four point zero zero one at the last row. Nothing here is a coincidence of the squaring rule: the same four columns for any continuous rule converge in the same way.

FIG 3A limit of infinitely many sums, as one subtraction

: the total between the two ends: by definition a limit of sums over thinning slices, which nobody wants to compute
: a height times a width, which is what every term in those sums was. The notation keeps the memory of the construction
: any rule whose rate is f, evaluated at the far end
: the same rule at the near end. Subtracting removes any constant that was added to F, so the choice of F is free

To use it, the work shifts entirely. Nobody divides anything into slices. The question becomes: what is this the rate of? The total of the doubling rule is the squaring rule because the rate of squaring is doubling, read backwards. A table of rates learned in the earlier lessons becomes a table of totals by being read in the other direction, which is why those lessons came first.

The constant, and why it is free

Suppose two rules both have the doubling rule as their rate. Their difference then has

a rate of zero everywhere, and something whose rate is always zero never changes, so

the difference is a constant. All the candidates are therefore the same curve at

different heights.

The lesson stops here

2 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 plan The contents

This 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

The Two Questions Are One Question

The one-line reason

The constant, and why it is free

2 more paragraphs to go

This is the reading half

Read alongside