Measure theory: they are the same picture.
this post was submitted on 28 Jul 2024
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unless you're analytically calculating that integral you're just summin'
They are the same if you make the multiplicative factor infinitely small. Then it's the differential you know from integrals.
The integral symbol evolved from an S for "sum".
And that was exactly how I named my inklings (Integrelle and Summatia)
Can someone splain
The sum approximates the integral.
The top symbol, Σ (uppercase Sigma), is used in math to denote a sum of a list of values. There is clear separation between the values in the list: two adjacent items in the list have no item in between them.
The bottom symbol, ∫ (long s), denotes an integral, which is kind of a sum over a continuous function. Any two different points of the function, no matter how close they are to each other, will have infinitely many points in between them.
For pedants: the function values don't have to be continuous, but the range of x over which the integral runs does have to be continuous. I regret nothing.
Oh cool, thanks. So is this like an anti-aliasing joke or something? Like "if you discretize a small number of pixels, Rick Astley will appear pixelated, but if you interpolate between them, the image will appear clearer?"
Not quite, I think it means the source material is continuous instead of discrete. No interpolation.
But honestly at this point we're reading too much into it.
You can integrate over arbitrary domains, not even the range needs to be continuous. You often see integrals not written as \int_a^b, but instead as \int_C where C is just a set
I still regret nothing.