# Geometric meaning of reversing the limits of an integral

• I
• mech-eng

#### mech-eng

With respect to operations, I understand why an integral is multiplied by -1 when its limits reversed. But integral is geometrically an area so reversing the limits would not be able to change neither how large is the area nor the shape of the area. Would you please explain changing the limits of an integral geometrically? Can we use the rectangles which are approximation to an area for this purpose?

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If ##F(x)## is a primitive of ##f(x)##, then $$\int_a^b f(x)\;dx = F(b)-F(a)$$ from which the change of sign when interchanging a and b is obvious.

Geometrically, the situation is that you get a 'negative area' when you go against the positive x-direction. Just like when you get a negative 'area' when ##f<0## in ##[a,b]##.

Easiest to see when looking at a rectangle: ##f(x) = c ##
Integral is (b-a)*c, area is |(b-a)| *|c|

mech-eng
Can reversing the limits geometrically mean "putting the front end of the region to the back, and back end to the front?" This would be a different function but with the same area since all the points on x-axis is constant.

Thank you.

Can reversing the limits geometrically mean "putting the front end of the region to the back, and back end to the front?" This would be a different function but with the same area since all the points on x-axis is constant.

Thank you.
If you like, yes. A volume in mathematics is an oriented quantity. E.g. a length from left to right is say L, then it is -L from right to left. The volume of a parallelepiped spanned by vectors ##v_1,\ldots,v_n## is the determinate of these vectors, or the norm of ##v_1\wedge\ldots\wedge v_n##. Either way, if we swap two vectors, then the sign of the determinant as well as of the wedge product changes. This is also true for integrals.

Volumes are oriented, or more precisely: volumes are the absolute values of an oriented quantity which can distinguish left and right.

mech-eng
Geometrically, the situation is that you get a 'negative area' when you go against the positive x-direction

This is an important point to me. Are Riemann sums defined based on the direction?

Thank you.

This is an important point to me. Are Riemann sums defined based on the direction?

Thank you.
That depends on what you call a Riemann sum. If you only consider the areas, then it is width times height and you have no orientation. But if you strictly follow the formulas, then height as well as length is oriented, simply because ##x_n-x_{n-1} \neq x_{n-1}-x_n## and the same for height.

That depends on what you call a Riemann sum. If you only consider the areas, then it is width times height and you have no orientation.

If we can have no orientation for Riemann sums, hence can we do so for an integral? So this could cause changing limits of an integral not affect the value of it.

Thank you.

If we can have no orientation for Riemann sums, hence can we do so for an integral? So this could cause changing limits of an integral not affect the value of it.

Thank you.
But this would complicate the entire calculus. E.g. ##\int_{-1}^1 x^3\,dx = 0## oriented, which is why we calculate ##|\int_{-1}^0 x^3\,dx|+ |\int_{0}^1 x^3\,dx| = \frac{1}{2}## if we actually want to know the size of the area under the curve. Now if you do this on the level of Riemann sums, then you would get something like ##\lim_{n \to \infty} \sum \,\left|\dfrac{i-1}{n} -\dfrac{i}{n}\right|\cdot \left|\left(\dfrac{x_{i-1}-x_i}{2}\right)^3\right|## which is much more complicated, esp. if the example is more complicated. It is easier to work with oriented volumes and decide what to do if the actual purpose is clear: antiderivative or absolute volume?

mech-eng
But this would complicate the entire calculus. E.g. ∫1−1x3dx=0∫−11x3dx=0\int_{-1}^1 x^3\,dx = 0 oriented, which is why we calculate |∫0−1x3dx|+|∫10x3dx|=12|∫−10x3dx|+|∫01x3dx|=12|\int_{-1}^0 x^3\,dx|+ |\int_{0}^1 x^3\,dx| = \frac{1}{2} if we actually want to know the size of the area under the curve.

Calculus is a very broad subject with numberless of applications in many different fields, and I am at the beginning of this very broad subject. Would you please basically explain what is the main point of orientation over only area?

Thank you.

Calculus is a very broad subject with numberless of applications in many different fields, and I am at the beginning of this very broad subject. Would you please basically explain what is the main point of orientation over only area?

Thank you.
I meant calculus in the sense of a framework for calculations, not as analysis. It is unfortunate that English doesn't distinguish calculus (frame) from calculus (analysis) or algebra (arithmetics, calculation) from algebra (theory of objects defined by one or two operations).

A length, area, volume is oriented, always. There is no point of making it oriented, it already is. To make it unoriented would require additional work. So the question is, should we only consider absolute values?

It matters, if you go left or right, simply because it matters whether you calculate 3-5 or 5-3. Your bank account is a volume! The absolute value is the exception because it doesn't matter whether you wallpaper clockwise or anticlockwise. But the purpose (bank account or wallpaper) defines whether orientation plays a role or not. It is not wise to ignore it from the start regardless of the goal.

mech-eng
I meant calculus in the sense of a framework for calculations, not as analysis. It is unfortunate that English doesn't distinguish calculus (frame) from calculus (analysis) or algebra (arithmetics, calculation) from algebra (theory of objects defined by one or two operations).

Oh, wow, that's an awesome statement! I think I'm on the brink of understanding that better, maybe a couple of steps away.

I think I understand the concept of algebra (arithmetic) from algebra (abstractions on operations).

How would you describe the difference between calculus (framework for calculations) from calculus (analysis)? What I had more or less formed in my mind was calculus as extension of algebra (ie transformations on polynomials, transcendentals, etc.., basically a bunch of useful rules to transform one function in another), and a second kind of calculus on abstractions, that seems complete elusive to me.

What calculus is in english, and here on PF is called analysis here. It is about analyzing functions and related topics. A calculus (with an article!) is a certain framework to solve or approach certain problems. E.g. the Lagrange multiplier to solve optimization problems could be named (and possibly is, I'm not sure) Lagrange calculus. However, as the Lagrangian is a fixed name in physics, which is also a calculus, this might cause confusion; or maybe not, they are closely related and basically the same thing. Predicate logic is a calculus, the Hilbert system is one, Landau symbols are a calculus, integration (alone, not as part of analysis as a whole) is a calculus. Short: a certain framework for certain calculations, not all in one.

I'm uncertain how to explain it better. Google translates the word I'm looking for with calculus and I have been told earlier here that calculus has also that meaning in english, too. So it is used for both in english, although the more common association seems to be analysis, and not a certain framework to do calculations in. Btw., there is no english version on Wikipedia of what I mean.

Same thing with algebra. Algebra is "calculation with letters" at school, but it is also, and this is named abstract algebra in english: group-, ring-, field-theory and the theory of algebras (as a specific vector space with multiplication) and modules. Here we have even three different meanings: school math, group theory and the algebra as a mathematical object. Instead of adding the adjective abstract to algebra, it should better be understood without, and algebra without this adjective is basically school math - just that algebra sounds more elaborated. E.g. ##x^2+4x+4## and doing some algebra we find ##x=-2## should in my opinion better be phrased as and doing some school math or basic math. Doing some algebra would read: ##x^2+4x+4## splits over ##\mathbb{Q}## with the double zero at ##-2 \in \mathbb{Q}## which shows, that it is not irreducible. That would be doing some algebra.

mech-eng, sysprog and BvU
I see... so, in your mind, how do you distinguish abstract algebra from calculus? One could say that day-to-day algebra (like 2+2=2*2) is a special type of abstract algebra (like, Lie Groups), and abstract algebra is a special type of calculus (something I'm pretty sure I don't have enough neurons to understand)?

PS: I think you posted that very well! I suspect that as day-to-day algebra and more abstract algebras share the same name and notation, that drives a lot of people very confused between the two, and that generates eternal questions like "how can ##i^2 = -1##... that's impossible!... it doesn't exist... crazy dudes!". I only understood the real point of ##i## once I started to read here.

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I see... so, in your mind, how do you distinguish abstract algebra from calculus? You'd say that day-to-day algebra (like 2+2=2*2) is a special type of abstract algebra (like, Lie Groups), and abstract algebra is a special type of calculus?
No. Algebra and calculus are both mathematics and they have touching points, but are basically two different things. They only share the property, that they are ambiguously used.

algebra - doing basic arithmetic operations
(abstract) algebra - e.g. group theory, Galois theory ; here: algebra
algebra - a vector space with a bilinear multiplication

calculus - here: analysis
calculus - a specific framework in which certain statements can be formulated and calculated

I will not change common usage, so it is senseless to suggest anything else. One has to live with this ambiguity. Don't forget that the discussion about it was only a consequence of my different usage of the word calculus, namely as the framework for Riemann integration; not as the very wide field it means as well.

sysprog
I see; thank you!

Here we have even three different meanings: school math, group theory and the algebra as a mathematical object. Instead of adding the adjective abstract to algebra, it should better be understood without, and algebra without this adjective is basically school math - just that algebra sounds more elaborated. E.g. ##x^2+4x+4## and doing some algebra we find ##x=-2## should in my opinion better be phrased as and doing some school math or basic math.
You're on the wrong side of history here. Solving an even simpler equation such as 2x - 3 = 1 would have been considered algebra by al-Khwarizmi, back in about 830 AD. For the next thousand years, algebra was synonymous with the idea of solving equations. I know that you conflate arithmetic and algebra (and we've had this conversation before), but in my view, arithmetic is exclusively associated with the so-called arithmetic operations -- addition, subtraction, multiplication, and division, without letters being used for variables. IOW, in my view, basic math and arithmetic are pretty much the same.
fresh_42 said:
Doing some algebra would read: ##x^2+4x+4## splits over ##\mathbb{Q}## with the double zero at ##-2 \in \mathbb{Q}## which shows, that it is not irreducible. That would be doing some algebra.
I would categorize this as Modern Algebra (i.e., concepts introduced only in the past 100 +-40 years.

I know the arabic origin of the word. However, language evolves, so why shouldn't it be adjusted to modern usage? The more as we have the term Diophantine equations for these kind of mathematics.

https://www.etymonline.com/search?q=letter says that to letter was used as a synonym for to instruct. Langauge changes.

Edit: I'm only saying that a distinction between a calculus and analysis is more convenient than to use the same word twice, and that algebra is even more ambiguous. But I just recognized that we run into the next problem by the use of analysis instead. (We distinguish Analysis (= calculus) and Analyse (= analyse something).)

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This is fascinating discussion!

I know the arabic origin of the word. However, language evolves, so why shouldn't it be adjusted to modern usage?
My objection was to the conflation of algebra with "school math," or worse, with arithmetic. Leaving off discussion of "an algebra or an algebra over a field," which has a very precise meaning, I don't see the need to raise the confusion level by using a new term for something that's been around for a very long time. I'm not at all confused when the conversation is about algebra versus abstract/modern algebra.
The more as we have the term Diophantine equations for these kind of mathematics.
Well, not really, since Diophantine equations are equations with integer solutions. At least here in the US, these kinds of equations are studied typically in a class on Number Theory, which is university-level.

fresh_42 said:
Edit: I'm only saying that a distinction between a calculus and analysis is more convenient than to use the same word twice, and that algebra is even more ambiguous. But I just recognized that we run into the next problem by the use of analysis instead. (We distinguish Analysis (= calculus) and Analyse (= analyse something).)
We in the US also distinguish between calculus (we don't usually call it a calculus) and analysis, with analysis involving a lot of proofs with sequences, as well as measure theory and so on.

It's easy to distinguish analysis from analyze, if only because the former is a noun and the latter is a verb, with the mathematical context of analysis being as I described above.

sysprog
fresh_42 said:
https://www.etymonline.com/search?q=letter says that to letter was used as a synonym for to instruct. Langauge changes.
That's another example of the English penchant for substitution of a specific for something properly more general. As a literary trope it's metonymy. An educated man may be called a 'man of letters'. In the US, we sometimes say a person 'schools' someone, when he educates someone, including outside of the context of a school.

We in the US also distinguish between calculus (we don't usually call it a calculus) and analysis, with analysis involving a lot of proofs with sequences, as well as measure theory and so on.
I was looking for the english counterpart of Kalkül. Google translates it with calculus, but there is no english Wikipedia version for Kalkül, and chrome translates it with calculus again. @WWGD told me - I think it was in General Discussions but I couldn't find it - that calculus is used in the sense of Kalkül in english, too, i.e. as a calculation machinery for a specific goal as in my examples in post #12. The Riemann calculus of integration is not the same as if I just say calculus. The former is only a small part of the latter; and the calculus of first order logic is something completely different than calculus.

I agree with you on the algebra example; just saying it's ambiguous, or context sensitive if you like this better.

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Mark44 said:
We in the US also distinguish between calculus (we don't usually call it a calculus) and analysis, with analysis involving a lot of proofs with sequences, as well as measure theory and so on.
I was looking for the english counterpart of Kalkül. Google translates it with calculus, but there is no english Wikipedia version for Kalkül, and chrome translates it with calculus again. @WWGD told me - I think it was in General Discussions but I couldn't find it - that calculus is used in the sense of Kalkül in english, too, i.e. as a calculation machinery for a specific goal as in my examples in post #12. The Riemann calculus of integration is not the same as if I just say calculus. The former is only a small part of the latter; and the calculus of first order logic is something completely different than calculus.

I agree with you on the algebra example; just saying it's ambiguous. or context sensitive if you like this better.
Partly because Latin and Greek are 'perfect' languages, ('perfect' here being used the sense of meaning per-fect, or completed or 'past', as in the Perfect Tense of the Latin verb), incipiators of emergent disciplines have tended to prefer those ancient languages over in-flux modern languages when naming such disciplines or their intellectionary artifacts.

A 'calculus' in Latin is literally a small stone (or, by extension, any bead or bead-like object such as might be used on an abacus) such as one might use for counting.

In the tradition of using a specific to substitute for something more general, 'calculus', or 'a calculus' becomes any procedural or conceptual apparatus used in 'calculation', or, more Germanically, 'reckoning'. Similarly, a British school child may refer to the entirety of mathematics as 'sums', quite comfortably, while knowing better.

Physicians may call kidney stones 'renal calculi', but in general they would prefer to say that the minerals involved are 'lithiating' or perhaps that 'biliary salts are in macroscopically observable lithiasis' rather than that the minerals are 'calculating', i.e. becoming pebble-like in a crystalline accretional process.

When someone wants to move from generality back to specificity, a qualifying term may be prepended to 'calculus', e.g. 'differential calculus', 'integral calculus', 'single-variable calculus', 'propositional calculus', etc., and someone may refer collectively to all such disciplines as 'calculi'.

weirdoguy and fresh_42
Physicians may call kidney stones 'renal calculi'
And dental hygienists remove calculus from your teeth, also in the "small stones" sense of the word.

And dental hygienists remove calculus from your teeth, also in the "small stones" sense of the word.
Right. The root word is used rather promiscuously. A dental professional is more apt to refer to 'the calculus', in the sense of 'calculus' as 'stuff' rather than to 'a calculus' as a thing (or in the plural, to 'calculi' as things, as urologists might) , while a mathematician referring to 'the calculus' would most likely be referring to the marvelous methods invented by Newton and Leibniz, and decried by Berkely for the associated inconsistency in the treatment of an infinitesimal as non-zero when it is one of an infinitude of them being summed, and as zero when it is individuated and disregarded at the asymptotic limit.

But this would complicate the entire calculus. E.g. ∫1−1x3dx=0∫−11x3dx=0\int_{-1}^1 x^3\,dx = 0 oriented, which is why we calculate |∫0−1x3dx|+|∫10x3dx|=12|∫−10x3dx|+|∫01x3dx|=12|\int_{-1}^0 x^3\,dx|+ |\int_{0}^1 x^3\,dx| = \frac{1}{2} if we actually want to know the size of the area under the curve.

A length, area, volume is oriented, always There is no point of making it oriented, it already is. To make it unoriented would require additional work. So the question is, should we only consider absolute values?
Would you please explain if "oriented" here means "have a direction", such as "vectors" do?

Thank you.

Would you please explain if "oriented" here means "have a direction", such as "vectors" do?

Thank you.
Yes, volumes have a direction, but the usual term is orientation.
Length: left to right versus right to left on the road
Area: clockwise versus anti-clockwise around the block
Volume: left hand rule versus right hand rule to order the dimensions

This is also true in higher dimensions, but then I do not have comparisons anymore; except to say that volume is a determinant and these are alternating and thus oriented: If we switch neighboring columns or neighboring rows, then the sign of a determinant changes, only the sign, not its absolute value.

Yes, volumes have a direction, but the usual term is orientation.
Length: left to right versus right to left on the road
Area: clockwise versus anti-clockwise around the block
Volume: left hand rule versus right hand rule to order the dimension

I am confused now. So are "volumes" and "integrals" vector quantities such as "velocity", "force" etc?

Thank you.

I am confused now. So are "volumes" and "integrals" vector quantities such as "velocity", "force" etc?

Thank you.
You are right, we only consider the positive value a length, area or volume.

A volume is a norm, so it is automatically made positive. But calculated is a quantity which can change signs. The absolute value comes in with the word volume. It is made positive since we cannot buy ##-1## liter of water. But if we introduce a coordinate system to precisely say, where the cube of water is placed, then the cube which contains the liter has an orientation depending on whether our coordinate system is right- or left handed. If we integrate over this cube, then we calculate within this coordinate system and we get a negative or positive number at the end. Of course we only call the positive number a volume, but this is an extra step at the end of the calculation. The cube itself is oriented, the integral over the cube likewise. Volume is then the absolute value of this integral.

Sorry for confusion. Road, block and cube are oriented, length, area and volume are the absolute values of the content aka integral. I'm so used to consider it as possibly negative, because that's what the outcomes of calculations are, that I forget that the volume is the absolute value of this outcome. This is important to know if the quantity we compute changes orientation in the process, such as areas below and above the ##x-## axis during an integration.

@fresh_42: 1 stere ##= 1m^3 = 1,000l##, so, if we have, at the end of our 'stere of beer' party, no remaining beer, but still have the empty containers that held the thousand liters, we could recognize that of our original thousand-liter supply, we now have zero beer, just as if we'd had no beer to begin with, but we could also identify the magnitude of the empty containers by its 1 stere volumetric measure, and we could also distinguish it from full containers by use of a negative sign ...

Yes, the discussion has a bit of the old math joke, where a professor held a lecture and only three students came. All of a sudden all five stood up and left. "Damn", thought the professor, "I hope there will be two others arriving, so I can go home."

sysprog
In a not-so-dissimilar vein -- Physician says to man "you should stop smoking; you'll live longer". Man replies "but Doc, I'm 95 years old already". Doc says "yes, and if you hadn't smoked, you'd be much older".

fresh_42
Here's an example where the sign makes a difference.
If $f(x)$ represents a conservative force, where $x$ is a displacement,
then the signed area under the curve represents the signed-work done by the force.
I treat the points on the curve $(x,f(x))$ as a sequence of "states" in F-x space (as one would do in a Pressure-Volume diagram in thermodynamics).
Switching the limits corresponds to traversing those states in reverse,
yielding the opposite-sign of the work-done by that force.

Calculus allows for continuously changing object, standard Algebra does, not, it lives in a discrete realm. You cannot speak, e.g. of convergence or infinite sums within Algebra.

Also remember integrals may tepresent things other than areas. Consider velocity in x axis, time in other . Then integral represents total distance. If x-axis changes sign then this is just, as Fresh said, a change of orientation. as in Physics. Distance is measured in negative sense.

Calculus allows for continuously changing object, standard Algebra does, not, it lives in a discrete realm. You cannot speak, e.g. of convergence or infinite sums within Algebra.
A professor whose math class I took said something like "To see at once the beauty of the calculus, observe how it bridges between algebra and geometry". I liked that. His PhD was in Physics, but he was a family man, with the associated pressing obligations, and the job he was offered by the university was teaching math. He said he thought he'd have preferred to teach physics, but he was happy teaching math, in large part because it had so much that was so beautiful.