Cutting the interval into more pieces shrinks the window around the total. Doubling the pieces over and over gives a column of exact sums; the lab prints their errors against the exact total and the ratio of one error to the next. On a straight-line rate that ratio is exactly two. On a curved one it is close to two and is not two, and the lesson separates what the table shows from what it would take to prove.
Core ideas
Three ideas: what refining changes, how to measure progress with an exact ratio, and how not to over-read a table.
Doubling n halves h and keeps every old left end as a left end of the new partition. The new sum has twice as many terms, each with half the width, and each term uses a rate read closer to where the rate is actually being added.
The error of a sum is the sum minus the total. The ratio error(n) / error(2n) says how many times smaller the error became when the pieces doubled. A ratio of 2 is the error halving. The lab prints it as an exact fraction.
The sums 7/32, 35/128, 155/512 climb, and their errors shrink. The claim that they approach the total is stated here as a claim. The table is evidence for three values of n, and evidence is not an argument about all of them.
The material
A column of sums, two exact ratios, a rate for which the ratio is exactly two, and the line between demonstrating and proving.
The square rate t² on the interval from 0 to 1 has left sums that can be computed for any n. At n = 4 the left sum is 7/32; at n = 8 it is 35/128; at n = 16 it is 155/512. The lab's exact total is 1/3. Each sum is below it and each is closer than the one before.
The lab prints the error as the sum minus the total, so the errors of these low sums are negative: −11/96, −23/384, −47/1536. The ratio of one to the next is positive, because the signs cancel. The decimals are marked ≈ because they are rounded; 44/23 and 92/47 are the exact figures.
Each doubling cuts the error to a little more than half. If it were exactly half the ratio would be 2, and the ratios are 44/23 and 92/47, which are below 2 and creeping toward it. The reason has a simple core: the error is about half of the gap h·(f(b) − f(a)), and the gap halves exactly.
The ramp 2t on the interval from 0 to 2 is a straight line. For a straight line the true total is exactly the average of the left and right sums, so the left sum falls short by exactly half the gap: h·(f(b) − f(a))/2 = h·4/2 = 2h. At n = 4, h = 1/2 and the error is −1. At n = 8 it is −1/2. The ratio is exactly 2, every time.
For a polynomial rate on a fixed interval, the left sums Lₙ for n = 4, 8, 16, … approach a single number, and that number is the exact total change.
This is stated as a claim. The table above is the lab demonstrating it at three values of n for one rate, and the ramp's error 2h proves it for that one rate because 2h goes to zero as h does. For a rate like t² there is no such formula in sight, and nothing the lab prints is a proof for all n. The proof is the job of a first analysis course; the lab supplies the numbers that make the claim believable and specific.
The cubic preset is t³ on the interval from 0 to 2 with four pieces. The left sum is 9/4, the total is 4, and the error is −7/4. At n = 8 the sum is 49/16 and the error is −15/16.
The ratio is (7/4)/(15/16) = 28/15, about 1.86667, further from 2 than the square's 1.91304 at the same n. A rate that bends more needs thinner pieces before its error settles into halving, and n = 4 is still coarse for it.
Doubling the pieces is a measurement, and what it measures is a property of the rate. A ratio of exactly 2 identifies a rate whose left sum errs by a fixed multiple of h. A ratio near 2 and below it identifies one that errs by about that. Neither ratio is a verdict on the sum's correctness: every figure is exact, and the error is the method's.
Interactive
Left, right, trapezoid and midpoint sums as fractions
Method
Four or five numbers from one lab, and one sentence that says what they show.
Everything is compared against one exact total, so only the number of pieces changes between rows.
The sum tile and the error tile, as fractions. Keep the sign of the error: a left sum on a rising rate is below the total.
Set the pieces to 2n. The old left ends are still left ends, and the new sum uses all of them and the midpoints between.
The ratio tile does this. Compare it with 2: exactly 2 means the error halved; slightly under 2 means a little more than half is left.
Write what the table shows (the sums at these n, the ratios) in one sentence and what you believe it approaches in another. Do not merge them.
From the square at n = 4, double the pieces and find the new error and the ratio, by hand.
The squares 0, 1, 4, …, 49 add to 140, and the sum is 140/512. The ratio 44/23 is below 2, so the error at n = 8 is a little more than half of the error at n = 4. Had the ratio been exactly 2 the lab would print 2, and for t² it never does.
For a rehearsal, repeat the computation at n = 16. The supplied first move is that the sixteen left rates are 0, 1, 4, …, 225 over 256. You should land on 155/512 and a ratio of 92/47.
You should be able to compute left sums at n and 2n, find the error and the exact ratio from the lab's total, say why a straight-line rate gives exactly 2 and a curved one does not, and state in words what the table demonstrates and what it leaves unproved.
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