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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.
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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?! ...
Thanks. I meanwhile changed the link to the English one in my list of sources but forgot that one.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.
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.Typo? If Gabriel's horn has finite volume, we can certainly make one large enough to hold a gallon of paint.
Thank you! I corrected it.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.
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?
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.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.