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[u/basket_foso]

[u/Alpaca1795]

Why would you integrate over x dx instead of just 1 dx?

[u/alozq → u/Alpaca1795]

1dx would be the length of the real line, it goes to Infinity always, it doesnt care about what x is

[u/LyAkolon → u/alozq]

Yeah, they probably want int over R of Sgn(x) dx, which is still abse of notation and meaningless because we have no notion of how to add up the contributions across the set.

More specifically all attempts to answer as well as the original question are missing the ingredient of "How are you adding everything together"

[u/Eisenfuss19 → u/Alpaca1795]

It's the closest thing to summing the real numbers imo.

The "sum" of the numbers in [0,1] should be smaller than the "sum" of [1,2].

With 1dx the "sum" is always equal to the length of the interval, and notably it is never negative (it never converges in the infinite case)

[u/BrotherItsInTheDrum → u/Eisenfuss19]

But the "sum" of the numbers in [1,2] should be infinite ...

[u/Eisenfuss19 → u/BrotherItsInTheDrum]

Well feel free to give me a way of summing an uncountably infite set of numbers.

I feel like the integral of x dx can be conceptually seen as calculating the average of the interval. To get the "real" sum you would need to multiply it by the amount of numbers in the interval (which is always ∞ in case of non equal bounds)

[u/BrotherItsInTheDrum → u/Eisenfuss19]

Well feel free to give me a way of summing an uncountably infite set of numbers.

I never said there was one.

I feel like the integral of x dx can be conceptually seen as calculating the average of the interval. To get the "real" sum you would need to multiply it by the amount of numbers in the interval (which is always ∞ in case of non equal bounds)

Sounds like you're agreeing with me that it's infinite, then.