Can You Solve the Mathematical Mystery Behind the Art of Integration?

In summary, my school teacher used to say that "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase is, that differentiation is the calculation of a limit $$ f'(x)=\lim_{v\to 0} g(v) $$ for which we have many rules and theorems at hand. And if nothing else helps, we still can draw ##f(x)## and a tangent line. Geometric integration, however, is limited to rudimentary examples and even simple integrals such as the finite volume of Gabriel's horn with its infinite surface are hard to visualize. We cannot fill in a gallon of paint, but it takes infinitely many gallons to paint it?! The message, however
  • #1
fresh_42
Mentor
Insights Author
2023 Award
18,994
23,992
My school teacher used to say

"Everybody can differentiate, but it takes an artist to integrate."

The mathematical reason behind this phrase is, that differentiation is the calculation of a limit
$$
f'(x)=\lim_{v\to 0} g(v)
$$
for which we have many rules and theorems at hand. And if nothing else helps, we still can draw ##f(x)## and a tangent line. Geometric integration, however, is limited to rudimentary examples and even simple integrals such as the finite volume of Gabriel's horn with its infinite surface are hard to visualize. We cannot fill in a gallon of paint, but it takes infinitely many gallons to paint it?! ...

This article cannot replace the 1220 pages of the almanac Gradshteyn-Ryzhik but it tries on 1% of the pages to summarize the main techniques.

Continue reading...
 
Last edited:
  • Like
Likes Mondayman, ComplexVar89, Al-Layth and 6 others
Physics news on Phys.org
  • #3
jedishrfu said:
Here's the english wiki page on the Gradshteyn and Ryzhik book:

https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik

I remember using the CRC Math Tables book which was considerably smaller and then went the the Schaum's Outlines Mathematical Handbook of Formulas and Tables.
Thanks. I meanwhile changed the link to the English one in my list of sources but forgot that one.

More of them:

https://www.amazon.com/dp/B00OUR06EO/?tag=pfamazon01-20
https://www.amazon.com/dp/B005H841YQ/?tag=pfamazon01-20

Gradshteyn / Ryzhik is probably the classical one. I like the Russian origin in that case. Soviet mathematics was always very technical and emphasized its applications in engineering. And tables of integrals fit their expertise.
 
  • Like
Likes ChiralSuperfields
  • #5
kuruman said:
Typo? If Gabriel's horn has finite volume, we can certainly make one large enough to hold a gallon of paint.
No, and yes. The original one has volume ##\pi [l]< \text{ Gallon} [l]## so a gallon does not fit in. Yes, we can easily scale it to hold a gallon. The message, however, was its finity. I needed a finite upper bound in order to illustrate finite volume compared to its infinite surface. A gallon over 4 liters seemed ok, a) for ##\pi < 4## and b) for a quantity most of our readers are familiar with. Scaling would have missed the point. You can always scale it beyond any given finite upper bound.

But needing more paint to color it than you can fill in is absurd. If we fill in ##5## liters of paint, then it is colored from the inside, but cannot be colored from the outside although they are equally big?
 
  • Informative
Likes kuruman
  • #6
thank you so much
collecting useful integration techniques is my favourite pastime at the moment
 
  • Like
Likes fresh_42
  • #7
I think you mean you cannot fit in a gallon of paint: "you cannot fill in a gallon of paint" is not correct English, and correcting it to "you cannot fill it with a gallon of paint" means the opposite.

Probably clearer to rewrite it as "a gallon of paint will not fit inside yet is insufficient to paint its surface" or some such.
 
  • Like
Likes fresh_42
  • #8
pbuk said:
I think you mean you cannot fit in a gallon of paint: "you cannot fill in a gallon of paint" is not correct English, and correcting it to "you cannot fill it with a gallon of paint" means the opposite.

Probably clearer to rewrite it as "a gallon of paint will not fit inside yet is insufficient to paint its surface" or some such.
Thank you! I corrected it.
 
  • #9
fresh_42 said:
But needing more paint to color it than you can fill in is absurd. If we fill in ##5## liters of paint, then it is colored from the inside, but cannot be colored from the outside although they are equally big?

What if we take two horns, different sizes. Fill the larger one with finite amount of paint, and then put the smaller horn inside the bigger horn with paint... So we've painted infinite area with finite amount of paint...? Scary.
 
  • #10
Theia said:
What if we take two horns, different sizes. Fill the larger one with finite amount of paint, and then put the smaller horn inside the bigger horn with paint... So we've painted infinite area with finite amount of paint...? Scary.
It is already a paradox with one horn. The surface inside and outside are the same. So filling it with a finite amount of paint should have painted it inside, but doesn't.

The best explanation I have ever heard from a mathematician was at a colloquium about the Banach-Tarski paradox (the mathematician's way to double a ball of gold). I don't remember his name, but he said that it is not so much the axiom of choice that leads to the paradox but rather our limited understanding of the infinitely small such as a point. Paint is three-dimensional and finite, the surface of the horn is neither. Mathematical objects simply do not work this (the paint's) way. Nevertheless, the mathematical infinities are good enough to explain the world.
 
  • Like
Likes Theia
  • #11
Nice text.

1) It is hard to catch a point in the text where we switch from the Riemann integral to the Lebesgue one. It must be a lot of such points there.

2) The theorem of an infinite series integration looks nonstandard
the standard one is for example as follows
Screenshot_20230626_102146.png


3) Laplace, Euler and other great mathematicians of the 18-19 century would be laughing good if somebody had said to them that the differentiation under the integral was invented by Mr. Feynman.
The precise formulation is as follows (the Lebesgue integral version):

2.png


Inserted text is from G. B. Folland Real Analysis and...
 
Last edited:

1. What is integration?

Integration is a mathematical process that involves finding the area under a curve. It is essentially the reverse of differentiation, which is finding the slope of a curve.

2. Why is integration important?

Integration is important because it allows us to solve a variety of real-world problems, such as finding the distance traveled by an object, the volume of a shape, or the work done by a force. It is also a fundamental concept in many fields of science and engineering.

3. How do you solve an integration problem?

To solve an integration problem, you need to use integration techniques, such as substitution, integration by parts, or trigonometric substitution. You also need to have a good understanding of basic algebra and calculus.

4. What is the difference between definite and indefinite integration?

Definite integration involves finding the exact numerical value of the area under a curve within a specific interval, while indefinite integration involves finding the general formula for the area under a curve. In other words, definite integration gives a specific answer, while indefinite integration gives a function.

5. How can I improve my integration skills?

To improve your integration skills, it is important to practice regularly and work on a variety of problems. You can also seek help from a tutor or join a study group. Additionally, understanding the underlying concepts and formulas will make integration easier to grasp.

Similar threads

  • Topology and Analysis
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
6K
  • Math Proof Training and Practice
2
Replies
64
Views
12K
  • Quantum Physics
Replies
19
Views
3K
  • Math Proof Training and Practice
2
Replies
46
Views
10K
  • Math Proof Training and Practice
2
Replies
60
Views
8K
  • Math Proof Training and Practice
3
Replies
77
Views
10K
  • Math Proof Training and Practice
2
Replies
43
Views
9K
  • General Math
4
Replies
125
Views
16K
Back
Top