Integrals and negative area, or help Rob win an argument with his professor.

A question for the more math-oriented among us:
On my recent Calc test, this problem was asked:
True-False: (Integral symbol from 0 to 2) (x - x^3) dx represents the area under the curve y = x - x^3 from 0 to 2. Explain your answer.

My answer verbaitum:
It depends: since x - x^3 dips below the x axis, the integral does ** not ** represent the area bounded by x = 0, x = 2, y = 0, and y = x - x^3. However, if you consider the area under the x axis to have negative height (not only is there no area under the curve, there is negative area under the curve), then it does.

The teacher circled “It depends”, wrote “Not true”, and gave me half credit. Today, I argued that there was nothing intrinsically impossible about negative area. I pointed out that all formulas given for area would accept a negative height and return an answer without blowing up or being inconsistent. I pointed out that if you buried a 10 foot pole, although it would remain 10 feet long, the tip of the pole would not be 10 feet high. Finally, I pointed out since we were dealing with infinite sums and complex numbers (concepts which had no analogue in the real world), so conceptualizing negative area shouldn’t be impossible.

So, am I right? Would any dopers with formal training in math care to support/refute me?

The thing is you are computing a “real” value, not a symbolic mathematical value. “Real” area cannot be negative.

I’d give you half credit as well. As worded, the phrase “under the curve” loses meaning when you dip below the x axis, since your negative area is really above the curve (or between the curve and the x-axis if you prefer). I’d say it would be more correct to state: “False. The intregral represents the algebraic sum of positive area between the curve and the x-axis where y > 0 (below the curve) and negative area between the curve and the x-axis where y < 0 (above the curve).”

Now wait a minute. IANAM, but it seems to me that you can have negative area. Say I have a ten acre farm. Nine acres are productive (above the x) and one acre for a house and outbuildings (below the x).
That’s 10acres - one acre = 9 acres. Nine acres make money, one doesn’t.

I would have given you full credit, because it sure seems to me like you know what’s what. However, I agree with your professor that the correct answer is “false”.

Had your teacher (or the textbook you were using) previously given you a precise definition of what they meant by the phrase “area under the curve” in a case like this where the curve dips below the x-axis? If not, I’d say the question was poorly worded.

I hadn’t specifically noticed it before, but as I look through the textbook I use to teach calculus, I see that it seems to avoid the phrase “area under the curve” altogether, instead using the “bounded by” kind of description you used in the first part of your answer.

Going just on what you’ve said, if I were in your teacher’s place, I would have given you full credit, while making a note to myself to reword the question next time around. The first part of your answer shows that you understand what’s going on, and if the second part is maybe a little unclear and oogly, so was the original question.

You obviously understand the concept. OTOH a true or false answer was expected, so it wouldn’t be fair to the other students to give you full-credit.

1/2 credit is ok


I get that I was unclear with my answer and shouldn’t get full credit (humbug), but my curiosity is piqued about negative area. If I was to tell you that a rectangle had base 2 and height -1, would it not have an area of -2 square units?

Well, metrics in general do not allow for negative distance. But I guess if you had a negative distance, a negative area would only make sense.

I agree with those who say that the question is poorly worded. Unless these words have been precisely defined in class or in the text, the terms “area” and, especially, “represents” are ambiguous[sup]*[/sup]. They have multiple valid interpretations. You explored some of these different interpretations and gave their respective correct responses. I would have given you extra credit.

*[sub]“Represents” is totally meaningless outside of some particular theoretical framework, but no applicable framework has been provided. Regarding negative area, there is such a thing as “signed area”. E.g. the determinant gives the signed area of a parallelepiped, depending on the order in which the generating edges are taken. In general, there exist signed measures (i.e. areas, volumes, etc.), and even complex-valued measures. See definitions 92 and 93 of this PDF[/sub]

  1. Definite integrals can be interpreted as area under the curve. This is not their definition. (Which is based on the inverse of derivative.)

  2. Definite integrals can be negative. Absolutely no argument is possible on this point. Any standard calc book will tell you this. In particular, if you reverse the range of the integral you get the same answer but reversed in sign.

  3. Therefore “negative area” is a perfectly acceptable shorthand for an definite integral (or part thereof) that comes out negative. I would give the OP full credit and “well done” on that answer.

This may not be standard, but in the text I learned from, definite integrals are defined in terms of limits of sums. Indefinite integrals are defined as antiderivatives, and the two are linked with the Fundamental Theorem of Calculus.

Distance is positive by definition.

Area is positive by definition.

The part your prof took off for is not because of any lack of understanding of calculus, however.

I think half credit is about right.

For area questions, I was always always taught to put magnitude bars around integrals, or a negative infront of the part of an integral that would yield a negative value, since area cannot be negative.

From someone who has taught calculus countless times … your answer is good, but you didn’t answer True or False. In these kinds of tests, it’s always better to clearly state T or F, then state the assumptions behind your answer. You should’ve stated False together with the first sentence in your answer. For your submitted answer, I also would’ve given you 1/2 credit. That said, unless your textbook defines “area under the curve” to be synonymous with “area bounded by the curve”, the question is clearly ambiguous. If it’s ambiguous, I would’ve thrown out the question. Believe me, I’ve done that a lot in my tests! That’s why I avoid T or F questions. There’s always a “It depends…”.

I may be wrong, but I suspect that the wording was that way for exactly this purpose–to make sure that the student is aware of area, and the possibility of a zero integral even though there is obviously “area” bounded by the curve.

Achernar was correct to object to this. Though I have seen books that define it this way, the standard approach in most modern Calculus books is to define the definite integral as a limit of Riemann sums. It’s a horribly unwieldy definition to work with directly when evaluating integrals, but it does make it reasonable to interpret the definite integral as an area, and it has the advantage that a function need not necessarily be continuous to be integrable. If it is continuous, the Fundamental Theorem of Calculus relates the definite integral, defined thusly, to the antiderivative.

Mathematicians have also developed more advanced definitions of the integral that agree with the “limit of Riemann sums” definition but which can also be applied to weirder functions. (This may be the kind of thing you have to be a mathematician to appreciate.)

Sometimes tests are badly phrased. I recently was asked by a friend the correct answer to a question which was structured like this:

Choose which one of the following is NOT TRUE:

A - The sun rises in the east (true)
B - Washington is the Capital of the USA (true)
C - An inch is 2.54 cm (true)
D - All of the above are true

A, B and C are true so they are all out but D is Also true and threfore does not meet the condition of being NOT TRUE. The question was very poorly phrased but one suspects the answer the teacher is looking for is D because you have to pick one. I told my friend to write an explanation and ask the teacher later but the teacher just dismissed her questions and said D was the correct answer. A jerk IMHO because the question should be clear and not confusing.