The Art of Integration

[fresh_42]

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...

 

[jedishrfu]

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.

 

[fresh_42]

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.

 

[kuruman]

We cannot fill in a gallon of paint, but it takes infinitely many gallons to paint it?! ...

Typo? If Gabriel's horn has finite volume, we can certainly make one large enough to hold a gallon of paint.

 

[fresh_42]

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?

 

[Al-Layth]

thank you so much
collecting useful integration techniques is my favourite pastime at the moment

 

[pbuk]

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.

 

[fresh_42]

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.

 

[Theia]

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.

 

[fresh_42]

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.

 

[wrobel]

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

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):

Inserted text is from G. B. Folland Real Analysis and...