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Comment by micheljansen | Hacker News Reader

They key ask being "without using calculus". I think my brain has been ruined by calculus in that I now consider approximating it as either an infinite stack of thin discs or as a pyramid with infinite sides to be intuitive explanations.

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I think the calculus-based explanation is the simplest one, and also easy to extend to an arbitrary number of dimensions.

If we look at the direction from the tip of the cone to the base, the volume is an integral of the area of the circles that are the cone sections. The radius of those circles grows linearly when the point is moving from the tip of the cone to its base. The area grows proportionally to a square of the distance from the tip. An integral of x^2 is x^3/3. Hence 1/3.

I think this is an elegant explanation

(x^3)/3 is so much faster for me to read and far harder to read incorrectly. Yes there's the memory trick for order of operations, but division symbols are so easy to get confused around that the extra clarity of the parenthesis should be standard for all non-trivial (one item per side) cases.

x^(3/3) would just be x though

It's my first thought too, but I never developed an intuitive understanding of why the integral of x^2 is (x^3)/3 (even though I know it is by heart).

Do you have intuitive understanding of the derivative rules? For example:

d/dx of x^3 = lim h->0 of ((x+h)^3 - x^3)/(h)

= lim h->0 of (x^3 + 3hx^2 + 3xh^2 + h^3 - x^3)/(h)

= lim h->0 of 3x^2 + 3xh + h^2

= 3x^2

So then if we started with (x^3)/3 the 3s would cancel and we’d get x^2. This tells us that the antiderivative of x^2 is (x^3)/3.

Or did you mean some other intuition? Such as why the fundamental theorem of calculus (that integration and differentiation are inverses of one another) is true?

Graphing y=x^0 -> y=x^1 and y=x^1 -> y=(1/2)(x^2) and doing math on the shapes (just simple squares and rectangles) makes those pretty intuitive. I just see it as an extension of that.

I'm the opposite, I never really understood calculus until I realised that integration is low pass filtering and differentiation is high pass filtering, and since I was in high school I've only ever converted calculus into R, C, Fin, and Fc, and solved the filter.

Differentiation is high pass filtering? How so? Is low pass filtering followed by high pass filtering the same as leaving a signal unchanged?

The answer to your question requires a lot of math but in my opinion, thinking about derivatives and integrals in terms of signal responses is not at all intuitive since your have to first interpret the signal in terms of its fourier transform (so a sum of different magnitude and frequency sinusoids) and then you have to think of the operations as phase shifts that are only applied to a certain set of frequencies (cosines and sines are 90 degrees out of phase and are the derivates and ontegrals of eachother). The integrator and differentiator will apply the phase shift to different sets of frequencies. What complicates things are that the magnitudes are also changed based on the frequency of the sinusoid. I think the person you are replying to happened to learn this stuff before actually doing calculus and to him it seemed intuitive. If you would like to learn more, any control systems textbook will be able to do the work a lot more justice than I can.

In image processing an edge detector like Sobel works (IIRC) by forward differencing. This is not quite differentiation but may help connect the dots a little bit. My guess is that some people may find this more intuitive.

In simple terms, every sin(wt) or cos(wt) term becomes wcos(wt) and -wsin(wt) when differentiated, so the higher the frequency, the higher the multiplier.

This is not what most people would call high pass filter, as there's no cutoff frequency etc, and the phases get out of whack quickly, but you can think of it in these terms if it helps.

If you call multiplying the Fourier transform by a linear function "high-pass filtering" and dividing by a linear function "low-pass filtering", then I understand what you are saying. But I'm not accustomed to calling those high-pass filters and low-pass filters. I think of high- and low-pass filters as multiplying the Fourier transform by a step function (ideally, or some close approximation of such).

I spent freshman year taking calculus and physics and just being mad at all of the science classes I’d taken so far.

All this 1/2 bullshit is just integration?

My first intuition is integrating a "triangle" using cylindrical coordinates.

Is "circular pyramid" intuitive, or is it calculus based? It works for me.

Is Cavalieri's principle (two volumes are equal if every cross section has equal area) considered calculus or not?

I'd say that this is an application of Fubini's theorem which is very clearly calculus (or analysis, not sure where the distinction is drawn in English).

calculus = introductory real analysis aimed at engineering and science students. Same material, different emphasis.

I would say "circular pyramid" is intuitive, but "an n-sided pyramid works just the same as a square pyramid" is not.

If instead of just a square pyramid, you demonstrated it for regular polygons with sides 3, 4, 5 - I wouldn't need any convincing that the relationship continued to infinity.

It's both.

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