Choosing vertical or horizontal strips, and integrating in the matching variable

0:00There are two ways to cut a region into strips, and both of them are correct. Stand the strips upright on the horizontal axis, each one running up to the curve, and add them along that axis between two vertical lines. Or lay them flat against the vertical axis, each one running across to the curve, and add them up that axis between two horizontal lines. Two set-ups. Two integrals. Equal standing.

0:32And nowhere, in the place where you first meet them, is there a rule for choosing between them. That gap is what this is about. Because the choice is not a matter of taste, and there is a way to find out which direction a region is asking for before you write anything down. Start from what a strip already is. A thin slice of the region. One of its two measurements is a thickness, and it is going to be made small. The other is a length, and it is read off the boundary.

1:07Multiply the two, add up all the slices, and let the thickness go to nothing. That is the whole construction, and notice what it does not say. It does not say which way round the slice lies. Nothing in it prefers upright to flat. The preference you have is a habit, picked up from every example you have ever been shown, and habits are not conditions. So turn one through a right angle, in place, and watch the labels follow it round.

1:40Before the turn: the thickness is measured along the horizontal axis, and the length is measured up to the curve. After the turn: the thickness is measured up the vertical axis, and the length is measured across to the curve. The two measurements have swapped which axis they belong to. That is the swap, and it is the easy half of the story. Here is the half that gets skipped. Cut ten different regions into the same number of strips both ways, and ask how thick a strip is.

2:16On seven of the ten you get two different thicknesses, one for each direction. On three you get one thickness both ways. The thickness belongs to the direction you chose. It does not belong to the region. The strip is not the only thing that turns. Upright strips are closed in by two vertical lines, and those two lines supply the limits. Flat strips are closed in by two horizontal lines, and those are different lines at different places, and they supply different limits.

2:53So changing direction is three changes at once. The variable that carries the thickness. The rule that supplies the length. And both limits. Not one change with two consequences. Three changes, and if you make one of them and forget the others you will get a number, and it will be wrong, and nothing in the working will look odd. That is worth measuring rather than warning about. Take those same ten regions. Turn the rule flat, correctly. Then leave the limits where the upright strips had them, and integrate.

3:33On three of the ten, the answer comes out right anyway. On three, it comes out wrong. And on four it cannot be worked out at all, because the sideways rule was never defined that far and simply returns nothing. So a limit left behind is caught two thirds of the time. Which sounds reassuring until you look at the three it lets through. The three it lets through are exactly the three regions whose two limits happen to be the same number.

4:06A quarter circle is one of them. It reaches the same distance along as it does up, so the upright limit and the flat limit are the same number, and leaving one behind changes nothing. It gets worse. On a quarter circle the rule you read up the region and the rule you read across it are the same rule. Across ten regions, that happens on exactly two, and both of the two are circles.

4:36So when a quarter circle is set up both ways, and the two answers agree, nothing has been demonstrated. The second integral is the first one written out again, with the letters exchanged and every number identical. It is the most natural region to demonstrate on, and it is the one region that cannot show you anything. Stretch that circle into an ellipse and the demonstration comes alive. Now the two directions do different work. Each set-up pulls a constant out in front, and the two constants are the two radii divided one way and then the other.

5:19Across six ellipses, the two constants are reciprocals every time. That is why the two set-ups can differ in every visible part and still land on the same number. And what survives inside the integral is different too. It is the circle on the along radius one way and the circle on the up radius the other, and on five of the six those are genuinely different numbers. So the constants are not cancelling a coincidence. They are carrying two different readings to the same place.

5:55Put the constant on the wrong way up and you can watch that fail. Six ellipses, and it is right on exactly one of them. The one is the ellipse whose two radii are equal. Which is to say, the circle again. None of that tells you how to choose. It only tells you that choosing badly is expensive. Here is the constraint, and it is the one thing nobody says out loud.

6:24A strip needs a length. To have a length, the boundary has to hand you one number at that place, and one only. If a cut across the region meets the boundary at two places, the strip has no length until you decide what to do about the second one. That is the test. Not which direction looks nicer. Not which one you are used to. Ask how many times a cut meets the boundary, and if the answer is one, that direction is available.

6:57You can answer that without rearranging anything, which matters, because rearranging is the step that hides the problem. Take the equation of the boundary exactly as it is written. Fix the cut. Now walk along the cut and watch the equation change sign. Every place the sign flips is a place the cut meets the curve. Every place the equation comes out exactly nought is a meeting too. No square roots taken. No choosing between plus and minus. Just counting.

7:34And the count really can come back nought, one or two. A level line clear of a circle meets it nowhere. The same line drawn inside one corner of that circle meets it once. Drawn across the whole circle, twice. Run that counting over ten regions, in both directions, at twenty-one places each. Four hundred and twenty cuts. Every single one meets the boundary exactly once, and not one of them meets it twice.

8:08So on all ten regions, both directions are available. Either set-up will work, and the two will agree. Which makes those ten regions completely useless for showing you when the choice matters. This is exactly the trap of demonstrating on a shape where both directions work. It proves the two agree. It cannot prove anything about choosing. So take something away and watch the count change. Every one of those ten regions sat in the corner where both coordinates are positive. Remove that restriction. Keep the same ten equations, and let the cuts run over the whole of each curve.

8:53The ten split three ways. Three of them are still single wherever you cut them. Their equations hand back one value in either direction, everywhere, and no corner is needed. Three are single one way and double the other. One direction is safe and the other is not. And four are double both ways. Those four are exactly the four closed curves: the circles and the ellipses. A closed curve cannot be single-valued in either direction, and that is a fact about being closed, not about being curved.

9:33Read that back and something uncomfortable falls out. For seven of the ten, it is the corner doing the work, not the equation. The rearrangement is legal because of where the region sits, not because of how the curve is written. You will see this passed over in a bracket. A curve gets rearranged for the other variable, a small question mark appears beside the step, and the question is never answered.

10:03The answer is one sentence, and it is the same sentence both times it is asked. The piece being measured lies where both coordinates are positive, so of the two values the rearrangement offers, the region has already chosen the positive one. Not the equation. The region. Here is a region where a cut really does meet twice, so you can see what it costs. A curve shaped like a bowl, with a level line laid across the top of it. The region is what sits between them, and it reaches across the vertical axis, so it is not tucked into any corner.

10:46Cut it upright, at twenty-one places: every cut meets the bowl once. That direction never has to ask. Cut it flat, at twenty-one places: every cut meets the bowl twice, once on each side of the axis. Now the flat strip has no length until you say what you mean. And there is a right answer: the length is the distance between the two meetings. Take it that way and the flat reading agrees with the upright one exactly.

11:18Take one of the two meetings and call it the length, which is the natural slip, and you get precisely half the area. The whole of the other side of the region has quietly gone missing. Which brings us to the thing you actually do first, before any of this. You read the region. A region arrives as a sentence, and every clause in that sentence is a piece of the set-up. This curve. That axis. That line at that height.

11:51When a description names the vertical axis as a boundary, it is telling you where the strips start. That is a structural instruction, not a decoration on the sentence. It is the easiest clause in the world to skim past, because the vertical axis feels like scenery. It is not scenery. It is one of the two lines that will become your limits. So take a region described exactly like that.

12:22A parabola lying on its side, whose up-coordinate squared is four times the along-coordinate. The vertical axis. And a level line three units up. Three clauses, and the region they shut in sits between the vertical axis on the left and the parabola on the right, from the bottom up to that line. Lay the strips flat and the set-up writes itself. A flat strip starts at the vertical axis and runs across to the parabola, so its length is the along-coordinate, and the equation hands that to you directly as a quarter of the up-coordinate squared.

13:04One curve. One value. Nothing to decide. Add those up the vertical axis from nought to three and the area is nine quarters. Now do the same region uprightly, because it can be done and it is worth seeing what it costs. An upright strip in that region does not stand on the horizontal axis. It stands on the parabola and rises to the level line. So its length is the difference of two boundaries: three, less the up-coordinate on the parabola.

13:40Add those along the horizontal axis, out to the place where the parabola meets the line, and the answer is nine quarters again. The same number. So this direction is not forbidden. Both work. But look at what each one asked of you. The flat strip read its length off one curve. The upright strip needed both boundaries, and needed you to notice that the region does not reach down to the horizontal axis at all.

14:12That is the difference the counting predicted. Flat cuts across that parabola meet it once each, all twenty-one of them. Upright cuts meet it twice each, all twenty-one, once the region is not there to choose for you. And here is the mistake that direction invites. You have an upright strip. You have a curve above you. Every example you have ever seen ran the strip from the horizontal axis up to the curve. So you run it from the horizontal axis up to the curve.

14:48That measures the region under the parabola, which is not the region you were given. It was on the other side of the curve. The number that comes out is nine halves. Nine halves is exactly twice nine quarters. Not a rounding slip, not an obviously silly answer, just a clean number of the right size that belongs to a different region. If you have a set of answers to choose from, a number like that is sitting in it waiting for you.

15:21There is a check that catches this every time, and it takes one line. The two regions, the one you were given and the one under the curve, are two pieces of the same box. That box is three high and nine quarters wide. Its area is twenty-seven quarters. Nine quarters plus nine halves is twenty-seven quarters. So the two pieces fit. Add your answer to the region you did not want, and if the total is not the box, one of the two is the wrong region.

15:56Ten seconds. And it does not care which mistake you made. It is tempting to shorten that into a rule of thumb. Out by a factor of two means you measured the other side. Do not. Here is why. Take nine different arcs across the same unit square, each one a different power of the input. For each, measure the piece under the arc and the piece beside it. On all nine, the two pieces fill the square. That part is general.

16:29But the ratio between them is decided by the power, and it is different for every one of the nine. Exactly one of the nine has the piece under the arc twice the piece beside it. The other eight are out by other amounts. So the factor of two belongs to that one curve. It is not the signature of the mistake, and looking for it will let the other eight through.

16:56Add the pieces and compare with the box. Do not learn the number. The box check is worth doing properly, because there is a way to get it wrong that feels right. Across ten regions: measure the piece under the arc, measure the leftover piece above it as its own separate sum, add them, and compare with the box the arc makes. All ten fit. Now do it the lazy way. Instead of measuring the leftover piece, reuse the reading you already have from the other direction, on the grounds that it is the same region seen sideways.

17:37It is the same region seen sideways. That is precisely why it is not the leftover piece. That version fits the box on one region out of ten, and fails on nine. The one it fits is the region with a straight lid, where the two pieces happen to be equal. Equal by accident, and the accident is that the boundary is a line. Back to the plus or minus, because it deserves a proper answer rather than a bracket.

18:10When you rearrange a conic for the other variable, you get a square root, and a square root offers two values. Take the other one and see what happens. Across four conic regions, every reading turns over in sign. The answer is the same size and negative. And of the eighty-four places that other root is read at, not one of them lies where both coordinates are positive. That is the whole answer. The negative root describes a different piece of the same curve, in a different quadrant, and the region you were handed is not there.

18:50So you are not choosing the positive root because roots are usually positive. You are choosing it because the region chose it before you arrived. One more thing about reading a description, and it is a small one that will save you a bad minute. A region can be described by a list of boundaries in which one of the items does almost nothing. Take a quarter disc of radius two, sitting in the positive corner. Its area is the circle constant itself, and read as strips it comes out exactly that, with the whole disc coming to four times it.

19:32Now suppose the description names a vertical line at the far right as one of the boundaries. At twenty-one places inside that region, the strip has a positive length. At that line, the strip has a length of exactly nought. The line touches the arc at a single point. It bounds the region along nothing at all. Meanwhile the horizontal axis, which really does bound the region along its whole floor, may not be named in the list at all. It arrives inside the words about which corner you are in.

20:10So describe the region to yourself. Do not read the list back. Now the good news, which is most of the time. When a cut meets the boundary once in both directions, the choice really is free, and you should take it. Both set-ups are correct, both give the same number, and the only thing left to compare is which integral you would rather write out. That is a legitimate reason. Pick the one whose antiderivative you can do in your head. Pick the one whose limits are nought and something.

20:47What you must not do is decide that first and check afterwards. The order is: read the boundaries, count the meetings, and only then choose on taste. Taste is the last criterion, not the first one. A warning about notation, because it costs students real time. The sideways set-up gets written with different letters in different places. The same object, named twice. It is worth being sure that is all it is. Take ten regions, rename the rule that supplies the length, and read every area again. Not one of the ten readings changes.

21:30And that is a result no test could ever have come out differently, because a letter is not something a sum can read. The sum sees numbers. So when the same construction turns up under a different letter, there is no difference to look for. Do not go looking for one. So here is the whole thing as a procedure. Read the region as a sentence, and turn each clause into a boundary. Notice which axes are named, because a named axis is where the strips start.

22:06Ask, for each direction, whether a cut meets the boundary once. Not whether you can rearrange the equation. Whether the answer comes back single. If exactly one direction survives, there was never a choice, and the region told you so before you wrote anything. If both survive, choose on whichever integral you would rather write, and then change all three things together: the variable, the rule and both limits. When you have a number, put it beside the region you did not measure, and see whether the two of them fill a box.

22:44None of that is a formula to remember. It is the reading you do before the formula, and it is the part that nobody writes down.