Regions below the axis, and why the sign has to be discarded before adding

0:00Here is a region between a curve and the horizontal axis, and here is the formula that measures it. The integral of the curve, from the left boundary to the right one. Strips standing upright, each one running from the axis up to the curve, added along the axis. It works. It has worked in every example you have been shown so far. Now slide the same region downward, without changing its shape at all, until it sits below the axis instead of above it.

0:34Same curve, same width, same region, same formula. And the number that comes out has changed sign. The region did not get smaller when it crossed. It certainly did not become less than nothing. So the formula has stopped telling the truth, and this is the one place it does. The sentence people repeat about this is that the area comes out negative. That sentence is where all the trouble starts, so let us fix it once, out loud, and then use the fixed one.

1:09An area is a size. A size is never negative. There is no region anywhere whose measurement is less than nothing. What is negative is the integral. The integral is not the area. It is a number that records both a size and which side of the axis the region sits on. Two different quantities, and for most of your life so far they have happened to coincide, because every region you were given sat above the axis.

1:41The moment one does not, they come apart, and everything difficult in this topic is a consequence of that one separation. Look at a single strip to see where the sign actually comes from. A strip below the axis has a height. The height is a length, it is drawn on the board, and it is positive, because lengths are. But the value of the curve at that place is negative, because the curve is below the axis there.

2:13The height and the value are different statements about the same strip, and the whole of this topic is keeping them apart. Take one region lying entirely below the axis and read all four hundred of its strips, one at a time. Four hundred readings, every single one of them negative. Four hundred sizes, every single one of them positive, and not one of the sizes negative. The sum adds the readings. Your eye adds the sizes. That is the disagreement, and it is not a subtlety, it is arithmetic.

2:52Start with the easy case, because it is the one where the usual repair is right. A curve that stays below the axis all the way across, from the left boundary to the right one. Nothing crosses. One sign throughout. The integral of this one reports minus eight thirds. The size of the region is eight thirds. So here you may take the number that comes out, drop its minus sign, and be finished. Put the bars round the whole integral and read off the size.

3:27That is the repair everybody learns, and on this region it is exactly right. The trouble is how wide that permission is, and the answer is: exactly as wide as the words one sign throughout. Not one step wider. Across eighteen different regions, measured properly, the size of the total and the total of the sizes agree on five of them and disagree on thirteen. The five are exactly the five whose curve never changes side anywhere inside the interval.

4:04And there is a second thing in that count worth more than the first. Where the two disagree, the size of the total is smaller. Never once larger. Not on any of the thirteen. So the sign-blind method does not scatter its errors in both directions. It fails in one direction only. It always under-reports, because pieces of opposite sign have cancelled each other before you ever looked. Here is the drawing that matters most in the whole topic.

4:39A curve that passes through the origin, dips below the axis, turns, and rises back across it, climbing away on the far side. The left boundary stands inside the part that is below. The right boundary closes the part that is above. Two pieces, filled, one on each side of the axis. The integral over the lower piece reports a negative number. The integral over the upper piece reports a positive one.

5:11And now the whole procedure, in the order it has to happen. Evaluate each piece separately. Take the size of each one, discarding its sign. Then, and only then, add the two sizes together. Take the sign off first. Add second. Reverse those two steps and you are computing something else entirely. There is a way this rule is usually written down, and it is worth looking at closely, because it is nearly right.

5:44It puts the bars round the first piece and leaves the second one bare. Size of the lower piece, plus the upper piece as it stands. On this drawing that is correct, and it is correct for a reason: you have already been told which of the two pieces is the negative one. You can see it. It is shaded on the page in front of you. In a question you have not seen before, nobody tells you.

6:12Test the two shapes of the rule against all eighteen regions. Bars round the first piece only gets twelve of the eighteen right and six of them wrong. Bars round every piece: eighteen right, none wrong. So carry away the general form, not the special one. Every piece gets bars. In the problems where the first-piece version happens to work, the general one works too, and it costs you nothing.

6:44Set the two procedures side by side, because students do not usually realise they are choosing between them. One: add the pieces up as they come, signs and all, and take the size of the answer at the end. Two: take the size of each piece first, and add the sizes. They look like the same instruction with the steps swapped. They are not the same instruction at all. The first one lets a piece below the axis subtract from a piece above it. The second one never lets that happen.

7:19And they can only give the same answer when there was nothing to cancel in the first place. How much does the first procedure actually lose? Not a little, and not predictably. Of those eighteen regions, on eleven of them the sign-blind number comes out at less than half the true size. And on six of the eighteen it comes out at nothing at all. Nothing at all is worth stopping on. That is not a wrong measurement. That is a method reporting that a region you can see, with your eyes, drawn on the board in front of you, has no size.

8:00It is the loudest failure available in this topic, and it is one line of arithmetic away from any student who does not split. So the pieces have to be found, which means finding every place the curve changes side inside the interval. Not guessed. Found. Set the curve equal to nothing and solve it, and keep only the solutions that lie between your two boundaries. Do that across all eighteen regions and it finds sixteen places where a curve changes side, and two more where a curve reaches the axis, touches it, and turns back without changing side.

8:44Those two are a different animal, and we will come back to them. Of the sixteen crossings, every single one lies strictly inside its interval. Not one of them is a boundary the question already handed you. That is the thing to notice. The crossing is never in the question. It is always something you have to go and get. Why the crossings, though, and not somewhere else convenient? Because a piece is only safe to measure when the curve keeps one sign right across it. Then, and only then, is the size of its integral the size of the region.

9:26Cut at the crossings and that is exactly what you get: across all eighteen regions, not one piece anywhere holds readings on both sides of the axis. Now cut the same eighteen regions at two places chosen for no reason at all, and count again. Thirteen of the eighteen are left holding a piece that still has both signs inside it. Cancellation goes on happening inside that piece, quietly, and the split has bought you nothing.

10:00And the five that survive the arbitrary cut are exactly the five that never changed side to begin with, so they were never in danger. The crossings are not a convention. They are the only places that work. Take a straight line, running up from lower left to upper right, with boundaries at minus one and one. It crosses the axis at minus two thirds. Not at minus one, not at one. At a place you have to solve for, sitting between them.

10:35So there are two pieces. A small one below the axis, from minus one across to minus two thirds. A large one above it, from there out to one. The lower piece has size one sixth. The upper piece has size twenty-five sixths. Take the sign off each, add them, and the region has size thirteen thirds. And the integral straight across, without splitting, reports four. Four is not thirteen thirds. It is short, as it is always short, by exactly twice the piece that got cancelled.

11:13That answer can be checked without any calculus at all, and this is the one place in the topic where it can, so it is worth doing. Both pieces are triangles, because the curve is a straight line. The lower triangle has base one third and height one. Base times height, halved, is one sixth. The upper triangle has base five thirds and height five. Base times height, halved, is twenty-five sixths.

11:44One sixth and twenty-five sixths. The same two numbers the integrals gave, exactly, with no calculus anywhere in sight. And to be sure that agreement means something, apply the same triangle formula to a region whose boundary is curved. It disagrees, both times, as it should. So the two routes agreeing is a genuine check, not a coincidence of arithmetic. When two independent methods land on the same number, the sign rule stops being an instruction somebody handed you and starts being something you know.

12:26Now the sharpest example available, and it is worth every second. A cosine across one full turn. It starts at height one, falls through the axis at a quarter turn, reaches its lowest point at the half turn, rises back through the axis at three quarters, and returns to height one at the end. Three pieces. Up, down, up. The first has size one. The middle one, the long one below the axis, has size two. The last has size one.

13:00Take the sign off each and add: the region has size four. Now integrate straight across the whole turn without splitting. The answer is nothing at all. Not four. Not two. Nothing. Look at the board. The region is there. It is shaded. It plainly has a size, and it is four. And the unsplit method has just reported that it has none, because the piece below cancelled the two pieces above it exactly.

13:32The same thing happens with a sine across a full turn, and it is worth doing once more, because the shape is different and the lesson is identical. The sine goes up first and down second. Two pieces, not three: one above the axis of size two, one below of size two. Take the sign off each and add, and the region has size four again. Integrate straight across, and again the answer is nothing at all.

14:04Nothing about that outcome is special to the cosine. It is what happens whenever what is above and what is below happen to balance, and there are a great many curves for which they do. Now a question of the kind you will actually be asked, with four answers offered and one of them to be chosen. The curve is a cube, and the boundaries are at minus two and one. It crosses the axis at nought, so there are two pieces.

14:35The piece below has size four. The piece above has size one quarter. So the region has size seventeen quarters. Seventeen quarters is the fourth answer offered, and it is the right one. But look at the second answer offered: minus fifteen quarters. That is exactly what the integral reports if you run straight across the crossing without splitting. And look at the third: fifteen quarters. That is what you get if you do run straight across, and then drop the minus sign at the end.

15:11Three of the four numbers on that list are accounted for. The right answer, and the two most natural wrong routes to it, each with its result sitting there waiting to be recognised. A student who integrates without splitting does not get a number that looks wrong. They get a number that is on the list. Here is a second one of the same kind, and it goes further. The curve is a variable multiplied by its own size, between minus one and one. Below the axis it is one rule, above it another, and the two meet at nought.

15:50Each piece has size one third, so the region has size two thirds, which is the third answer offered. The unsplit integral across the whole thing is nothing at all, and nothing at all is the first answer offered. And one piece taken on its own, by somebody who split correctly and then forgot to add the second half, is one third, which is the second answer offered. The right answer, the answer from not splitting, and the answer from splitting but stopping halfway. Three routes, three numbers, all three sitting in the list in front of you.

16:31None of that is a trap laid for you deliberately. It is simply that wrong methods produce reproducible wrong numbers, and a list of four answers has room for them. Now the case that sharpens the rule, and it is the one most people get backwards. Here is a V, made from a size: it comes down, meets the axis at a point, and goes back up. Boundaries wide on either side.

17:00Its formula changes at the corner. Below the corner it is one expression; above it, another. There is a genuine change of rule at that point, and it is visible. But the curve never goes below the axis. It touches the axis and turns straight back up. So this whole region needs no sign discarded anywhere. It is one piece. Its integral is nine, and nine is already the size, because there was never anything to cancel.

17:34By school geometry: two triangles, each with base three and height three, giving nine. The same answer, confirming it. Split where the curve crosses the axis. Not where the formula changes. Those are different instructions, and here they give different answers. How different? Count it, across all eighteen regions. Four of them are written as two cases, with a named place where the writing changes. Of those four places, exactly one is a place the curve crosses the axis. The other three are corners the curve turns at without crossing anything.

18:18And it is worse than that, in the useful direction. Sweep every one of a thousand and sixty-two places across all eighteen curves, looking for a corner: a place where the steepness on one side disagrees with the steepness on the other. That sweep finds three corners and nothing else. Three. And the one rule change that actually is a crossing is not among them, because that curve is perfectly smooth where its formula changes. You would never find it by looking for a corner.

18:53So looking for corners cannot find crossings, and finding a corner does not mean you have found one. The two ideas are simply unrelated, and the only test is the one that asks the curve where it changes side. One more trap, because it catches good students rather than careless ones. Over a stretch running the same distance either side of nought, a curve that turns upside down when the input does will always integrate to nothing at all.

19:28Six of the eighteen regions run over such a stretch. Three of those six turn upside down, and all three of them do integrate to nothing. Which is a true and useful fact, and then people draw the wrong conclusion from it: they decide the cancellation is a property of that kind of curve. It is not. Among the other three, the ones that do not turn upside down, one still integrates to nothing at all.

19:58Cancellation does not need symmetry. It only needs as much region below as above. Symmetry is one way to arrange that. It is not the only way, and it is not what you should be checking. There is a one-line version of the area rule that people carry into an examination. Area equals the integral of the curve, from the left boundary to the right one. No condition attached, nothing said about the sign, nothing about splitting, nothing about crossings.

20:35For a region sitting above the axis, that line is true. For thirteen of our eighteen regions, it is false, and on six of them it is not merely false but reports that the region does not exist. A one-line rule that is silent about its own condition is worse than no rule, because it will be applied confidently in exactly the cases it was never true for. If you carry one sentence away from this, do not carry that one. Carry the condition.

21:10So here is the whole thing, as four operations, in the order they have to be carried out. One. Find every place the curve crosses the axis strictly inside your interval, by setting the curve equal to nothing and solving. Crossings, not corners, and not the places where the formula changes. Two. Split the integral at each of those places, and nowhere else. Three. Evaluate each piece on its own, and write down what each one reports, minus signs and all.

21:46Four. Take the size of each piece, discarding its sign, and only then add them together. The order is the rule. Sizes first, addition second. If you add first and take the size afterwards, you have computed the other thing, and on thirteen out of eighteen regions the other thing is wrong. And if there are no crossings inside the interval, all of that collapses to the easy case, and you may drop a minus sign at the end without a second thought. That is not a different rule. It is this rule, with a step that had nothing to do.