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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?! ...

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...

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