

A Calculus Analogy: Integrals as Multiplication - bumbledraven
http://betterexplained.com/articles/a-calculus-analogy-integrals-as-multiplication/

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rms
Also check out _Elementary Calculus: An Infinitesimal Approach_ for a
mathematically rigorous course in infinitesimal calculus. I think it is much
more intuitive than typical limit calculus.

<http://www.math.wisc.edu/~keisler/calc.html>

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tel
If you already understand limit proofs of calculus go ahead and read the first
chapter and the epilogue to get a pretty complete picture on infinitesimal
calculus.

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lutorm
The "multiplying changing numbers" analogy made me think of "integrating
really is just calculating the average", ie integrating f(x)dx is like taking
the average of f(x) and multiplying by the size of the interval.

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christopherolah
I believe that the integral symbol with a dash through it is equivalent to an
integral divided by the interval...

Couldn't find any confirmation online, though.

Even if it isn't, it is a nice notation.

Maybe someone could confirm

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baddox
I don't get the "revelation" he's claiming that this is. When you're finding
work or distance, two examples he gave, you're still finding the area under a
curve. Sure, the axes will be representing time and velocity when calculating
distance, or force and distance while calculating work, but that is the exact
same thing as calculating area when the axes are used to represent distances
of width and height.

He's right that's it's akin to multiplication. In fact, you can find the areas
under trapezoids using very simple integrations, but he did nothing to
discount the fact that definite integrals are all about areas under the curve.

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leif
Seems to be helpful for physicists, not so much for mathematicians. A
physicist will look at an integral and say "oh, yeah, charge is field times
area, that makes sense!", while a mathematician isn't going to say the same
about measures. I think that's the "revelation" he was going for.

BTW, I always thought of convolution as multiplication. Anyone else feel
cheated by this guy for that reason?

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pixcavator
I feel that integration is more about addition than multiplication, and
differentiation is more about subtraction than division. Is it just me?

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pfedor
Not just you. I seems much more natural to me to think of an integral as
"continuous sum."

You can often describe the exact same thing using a discrete sum or an
integral, e.g., in a classical mechanics class you first consider a system of
N connected springs, and a number of properties (like, say lagrangian, energy
etc.) of the system will be expressed as sums with N terms. Then you take the
continuous limit, the connected springs become one string and voila, all sums
become integrals.

In quantum mechanics for some systems the set of all energy levels is discrete
(e.g., the harmonic oscillator), for some it's continuous (e.g., a free
particle), and often it is part-discrete part-continuous (a hydrogen atom.)
Many formulas will require you to take a sum for the discrete part and an
integral for the continuous part. Some quantum mechanics textbooks introduce a
special symbol, a capital sigma overlaying an integral symbol, which means
"integrate or add, whichever appropriate."

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aduric
Maybe it's just me, but I found Integrals more intuitive as a continuous "sum"
of probabilities in a distribution. So, if you want to marginalize a discrete
distribution, you just add all its components over the conditional variable.
If you want to marginalize a continuous distribution, you integrate.

I guess you could say that you're simply enumerating infinitesimal densities?

