Slicing a region into thin strips, and adding the strips with an integral

0:00You already know how to measure a rectangle, a triangle and a trapezium. Base times height, half of that, and the average of the two parallel sides times the distance between them. Then somebody draws a shape whose top is a curve, and there is no formula to write underneath it. The usual explanation for that is curvature: straight sides can be measured, curved ones cannot. It is a tidy story and it is false.

0:32Because you can also measure a circle. No straight side anywhere on it, and school geometry hands you its area without hesitating. So curvature is not what defeats the formulas. Not having a formula is what defeats the formulas. What follows is the machine that stops needing one -- and it is going to be taken apart carefully, because the line everybody memorises hides three separate claims behind two equals signs, and only one of the three costs anything at all.

1:09Here is the picture the whole argument rests on. A horizontal axis for the floor. Two vertical lines standing on it, one on the left and one on the right. And a curve running across the top, joining the tops of the two lines. That closed region is what we want the area of. Give the four corners letters, so we can talk about the region rather than point at it. P at the left foot, Q at the right foot. S directly above P where the curve begins, R directly above Q where it ends.

1:48One detail in that drawing is easy to skip and worth stopping for. The curve does not begin at S and end at R. It carries on past both, and the region is a piece cut out of a longer curve. That is not decoration. It says the region is a choice you made about a curve that was already there -- and the two vertical lines are where you made it.

2:14A second detail. P is clear of the origin. The left-hand wall is a general number, not nought. Almost every worked example you will ever see starts at nought, and it quietly teaches a habit rather than a method. Now stand a single thin strip inside the region, anywhere you like. Its foot sits on the floor. Its top is cut off by the curve. Its two sides are vertical, and it is narrow.

2:46Two measurements. Its height is however far it is from the floor up to the curve. Its width is a small change in the input -- small, and for now the same for every strip. And its area, says the argument, is height times width. That strip stands at no particular place. It is pinned only by its own position along the horizontal axis, somewhere between the two walls. Which is exactly what licenses one strip to stand for all of them.

3:20Then the whole region is all the strips, added up. And the line that says so is the one everybody writes down. That line has three expressions in it and two equals signs. First: the area of the region is the sum of the areas of the strips. Second: each strip's area is its height times its width. Third: the height is the curve's own value at that place. Written as one line, it reads as one fact. It is not one fact. It is three claims, and they do not cost the same.

3:58The third is a substitution -- it puts the rule of the curve where the word height was. The second is the rectangle formula, which you have had since you were small. The first is the only one with any content in it, the only one a picture is needed for, and the only one that can be false. So we are going to run all three. Not argue about them -- run them, on eleven different regions, and count.

4:28One promise before any of it, because everything after leans on it. Not a single area in this video comes from an anti derivative. Every one of them is a sum of strips: cut the stretch, read the curve's height once inside each strip, multiply by that strip's width, add. That is deliberate. Checking a claim about areas using the very machinery the claim exists to justify is not checking it.

5:00But a sum is only a reading where the width has stopped mattering. So every region is read twice, once with half as many strips, and the two answers have to sit on top of each other. Eleven regions, eleven settled readings, and each one lands on the number claimed for it. Move all eleven targets by a whole unit and the same test reports nought right and eleven wrong, which is what makes eleven out of eleven a reading rather than a formality.

5:34With one exception, and it is a useful one. The half circle does not settle at the strip count the other ten use. Its curve stands vertically where it meets the floor, and it needs ten times as many strips before the answer stops moving. Which is the first honest thing this method will tell you: it reports when it has not converged, instead of handing you a number anyway.

6:03Claim one. The region is exactly what its strips come to. Nothing lost between them, nothing counted twice. Here is how to run it. Cut each region into four pieces, of four genuinely different widths. Read each piece on its own, with its own strips. Add the four readings. Then read the whole region separately, in one go, with its own strips over the whole stretch. Those are two different pieces of arithmetic arriving at the same place. On all eleven regions they agree. Nought miss, nought unreadable.

6:44Now break it, in the two ways it can break. Slide each piece so that it begins a little before the one before it ended. The pieces now overlap, and part of the region is counted twice. Nought of the eleven survive. Slide them the other way, so each begins a little after the one before ended. Now there are gaps, and part of the region is not counted at all. Nought of the eleven survive that either.

7:14So claim one is not free. It can be wrong, there are exactly two ways for it to be wrong, and both of them are visible in a count. This is what the picture is for. Not to prove anything -- to make it obvious that the strips are laid edge to edge with no overlap and no gap. Claim three -- the height is the curve's value there -- looks like it costs nothing. So let us find the height some other way, and see whether the two agree.

7:47Stand at a place between the two walls. Start on the floor and climb the vertical line, halving your way upward, until you meet the curve. Where you arrive is the strip's height, found rather than looked up. Twenty-one places on each of eleven regions. Two hundred and thirty-one climbs. Two hundred and thirty-one agreements, nought misses. Which is a strange result to celebrate, so here is the version that is not free.

8:19A strip has two vertical sides, and therefore two heights: one where it begins and one where it ends. The line writes down one. Climb to the height of the other edge instead, one strip along. Now only twenty agree, two hundred disagree, and eleven cannot even be asked -- one for each region, because at the last place the other edge would stand past the right-hand wall. And the twenty that survive are all from one region: the one with a flat lid, where a strip's two edges are the same height and there was never anything to choose between them.

9:00So the substitution is free. Choosing which of the strip's two heights to substitute is not. That is the whole reason the total is an integral instead of a multiplication. Cut a region into forty strips. The forty widths hold exactly one value -- the widths are all the same, on all eleven regions. The forty heights hold forty different values on eight of the regions. On two of them they hold twenty, because those two curves are mirror images of themselves about the middle of their own stretch, so the heights pair off.

9:41And on one region the forty heights hold a single value. That is the flat lid. Try the multiplication anyway: take the height at the left-hand wall, multiply by the whole width, and see how often that is the area. It is right on exactly one of the eleven -- the flat-lidded one -- and wrong on the other ten. One factor holds still and the other one moves. When both hold still, you multiply. When one moves, you have to add.

10:15Which leaves claim two, and claim two is the one that looks harmless and is not true. A strip with a curved top is not a rectangle. Its lid slopes. Writing height times width for it is writing down the area of something else. That step is usually passed over in silence, and it is the only place in the whole argument where something false is written on purpose. What rescues it is not that the error is small. Small is not an argument. What rescues it is what the error does when the strip narrows.

10:53So measure it. Seven curved regions, at eight strips, sixteen, thirty-two and sixty-four. For each width, find the worst single strip -- the one whose rectangle is furthest from the true sliver. Take the height in the middle of the strip and that worst error falls by at least six times every time the width halves. Twenty-one steps out of twenty-one. Take it crudely, at the strip's left edge, and it still falls by at least three times, in all twenty-one.

11:28But shrinking is not the point either. Here is the point. Measure the error against the width of the strip it belongs to. That ratio falls too -- at every one of the twenty-one steps, both ways of choosing the height. The error in one strip shrinks faster than the strip does. That sentence is the entire justification, and it is the reason this is a derivation and not a fudge.

11:59Add the strips up and the same thing happens to the total. With the height taken in the middle, the total's distance from the region's own reading falls by at least three times at every halving -- twenty-one out of twenty-one. With the crude left-edge height it still falls substantially at every one of the twenty-one. So the answer is not an estimate. The rectangle is an approximation; the limit of the rectangles is not.

12:30That is worth saying plainly, because a lot of people leave this topic believing every area they compute is approximate. It is exact. The approximation lives inside the limit and does not come out the other side. And to be sure that count is measuring something, freeze the height. Take the height at the left-hand wall of the whole region and use that same number for every strip. Now the total's error does not fall by even a tenth at any of the twenty-one steps. It just sits there.

13:07One place it does not sit there: the flat-lidded region, where that frozen multiplication is exactly right at every strip count. Which it should be. That is the one region where the height really is a fixed number. There is one more thing hidden in the notation, and it is worth digging out. The line writes a single symbol for the width and uses it for every strip. Which only makes sense if all the strips are the same width -- and nothing anywhere says so out loud.

13:43Is that a condition of the argument, or a convenience of the writing? Run it and find out. Cut each of the seven curved regions into strips whose widths are all different. At eleven strips, eleven different widths, against the one width an even cut gives. Refine, and the error falls by at least two and a half times at every step -- twenty-one out of twenty-one -- and all seven of the finest readings land within a thousandth of the region's own.

14:16So no. Equal widths are a convenience. The argument does not need them. But something does have to shrink, and here is what. Cut so that the widest strip is never allowed to shrink -- leave a quarter of the stretch as one single strip, however finely you cut the rest. Now nought of the twenty-one steps improve by even that factor, and all seven readings have stalled. One of the seven happens to land within a thousandth of the right answer anyway, on the gentlest curve of the seven, which is what a stopped clock looks like rather than what a method looks like.

15:00The condition is not that the strips match. The condition is that the widest one closes. Now the two vertical lines, which are the easiest thing in the picture to mistake for decoration. They are not where you stopped drawing. They are the limits of the sum, and every number in this video depends on them. Walk the right-hand wall across a region, through twenty-one places. Every one of the eleven regions hands back twenty-one different numbers, and every one of the twenty steps between them goes up -- because a wall that has moved right has more region behind it.

15:43Walk the left-hand wall instead and you again get twenty-one different numbers, and this time not one of the twenty steps goes up. Move either line and the number moves. That is the whole of it. Which brings the last distinction, and it is the one students lose most often. An integral with no limits attached to it returns a family of functions -- all the same shape, sliding up and down by a constant. An integral with two limits attached returns a number.

16:19Carry forty-one different constants through the working. At any one place inside the region they give forty-one different numbers, so those really are forty-one different functions. Now take the difference between the two walls. One number, on every one of the eleven regions. Whatever was added at the right-hand wall was added at the left-hand one too, and it has gone. An area with a constant of integration attached to it is not an area. It is a sign that the two lines were read as decoration.

16:58Two conditions ride along with this, and only one of them usually gets written down. The one that does: the right-hand wall has to stand to the right of the left-hand one. Name them the other way round and every one of the eleven regions comes back negative. Nought positive. And each of those eleven is just the region's own reading with its sign turned over. An area is not allowed to be negative, so that condition is doing real work rather than being tidy.

17:34The other condition is not written down anywhere, and it is the more dangerous one. The curve has to stay above the floor. On all eleven regions here, the sum of the strips and the sum of their heights taken as sizes are the same number, so the distinction never shows itself. Now put the curve below the floor for part of the stretch. A cube across a balanced stretch. A straight line crossing the floor in the middle. A full turn of the sine.

18:08All three of those regions have plenty of area. All three of those integrals read nought -- the same single number for all three, while the three regions are plainly not the same size. Below the floor the strips come out negative, and the sum stops being an area and becomes a difference. That is not a flaw in the method. It is the method telling you the truth about what you asked for.

18:37Back to the shapes with formulas, now that there is something to test them with. Read every region as one single strip, with its lid drawn as the straight line between its two ends. That is the trapezium -- and the rectangle and the triangle are trapeziums with one side or another made special. It is right on all three of the straight-lidded regions, which is no surprise. On the eight curved ones it is right on one, and wrong on seven.

19:12That one is worth a moment, because it is a trap. It is not right because its lid is straight. It is right because that curve rises as far above the chord across one half of the stretch as it falls below it across the other, and the two errors cancel exactly. A method that is right once is not a method. It is a coincidence you have not looked at closely enough.

19:39And the circle keeps its formula. Half a circle of radius two is a half turn's worth of squares of the radius, and that lands exactly on the region's own sum of strips. Read the same half circle as one trapezium, using its chord, and it comes back wrong. So the honest answer to where school geometry stops is: wherever nobody has already done the work for you. One more thing the drawing was telling you, back at the start.

20:13The curve runs past both walls. So does the region depend on how much of the curve you were given? Widen the stretch each curve is defined on, well past the region on both sides, and leave the two walls exactly where they stand. Ten of the eleven can be widened, and all ten return the same number. Nought moved. And those ten widened curves really do answer outside the region, at places where the ones they were made from return nothing at all -- so the widening was real.

20:50The eleventh cannot be widened at all. Its curve is a half circle and it ends exactly where the region does. The region is a cut. The curve does not know where you cut it, and the answer does not either. This idea is much older than the notation for it. Long before anybody wrote an integral sign, the move was this: shut the shape you cannot measure in between two shapes you can, and then close the gap.

21:22Draw a many-sided figure inside a circle, and another one outside it. Both are made of triangles, so both can be measured with nothing but school geometry. At six sides, twelve, twenty-four, forty-eight, ninety-six and a hundred and ninety-two: the inside figure grows at every one of the five steps, the outside figure shrinks at every one of the five, and the circle's own number sits between them all six times.

21:55At six sides the two figures are more than four fifths apart. By a hundred and ninety-two they are under a thousandth apart. That number for the circle was never computed from the polygons, by the way. It was found separately, by halving. So the trapping is a reading and not an arrangement. Two thousand years later two people arrived at the same place from opposite directions: one treating it as the reverse of finding a tangent, the other as a sum of small pieces. It is the second of those that you have just been doing, and the elongated S was chosen because it is the first letter of sum.

22:42So what is actually in that one line. One claim with content: the strips fill the region, edge to edge, nothing lost and nothing double counted. Break it either way and every region fails. One claim that is false on purpose and rescued by a limit: the strip is a rectangle. Its error shrinks faster than the strip does, at every step, both ways of choosing the height -- and the total that comes out the other side is exact, not approximate.

23:17One substitution that costs nothing: the height is the curve's value there. Free -- until you have to decide which of the strip's two heights you meant. Two conditions, one of them written down: the right wall to the right of the left, and the curve above the floor. And two vertical lines that are not decoration, holding the difference between a number and a family of functions. None of that is harder than the formula. It is just the formula with its parts separated, so that you can say which part you assumed and which part you substituted.

23:57Which is the difference between remembering a line and being able to rebuild it.