Fundamental Theorem of Calculus
by dalvrosa on 4/23/2026, 1:09:56 AM
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/
Comments
by: EdwardDiego
> This post introduces the Riemann integral<p>Sweet! I'm keen to learn about the basic fundamentals of calculus!<p>> For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m<sub>k</sub> and M<sub>k</sub> denote the infimum and supremum of f on that subinterval...<p>Okay, guess it wasn't the kind of introduction I had assumed/hoped.<p>Very cool maths rendering though.<p>As someone who never passed high school or got a degree thanks to untreated ADHD, if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.
4/23/2026, 11:49:23 AM
by: mchinen
I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions.<p>I feel similar about the trace of a matrix being equal to the sum of eigenvalues.<p>Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.
4/23/2026, 11:05:03 AM
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4/23/2026, 11:39:50 AM
by: emacdona
> f is Riemann integrable iff it is bounded and continuous almost everywhere.<p>FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.<p>I mention that because when I read the first footnote, I thought this was a mistake:<p>> boundedness alone ensures the subinterval infima and suprema are finite.<p>But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).<p>Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.<p>If anyone wants a rabbit hole to go down:<p>Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)<p>Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]<p>[1] One of my favorite functions. It seems its purpose in life is to serve as a counter example. <a href="https://en.wikipedia.org/wiki/Dirichlet_function" rel="nofollow">https://en.wikipedia.org/wiki/Dirichlet_function</a><p>[2] <a href="https://en.wikipedia.org/wiki/Lebesgue_integral" rel="nofollow">https://en.wikipedia.org/wiki/Lebesgue_integral</a>
4/23/2026, 11:03:27 AM
by: mellosouls
<a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow">https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculu...</a>
4/23/2026, 10:21:26 AM
by: bikrampanda
What is the font used on the site?
4/23/2026, 10:13:47 AM
by: shmoil
Good job, David. Have a lollipop. Now learn & write up the proof that the Henstock-Kurzweil integral integrates _every_ derivative. This is what we had in my calculus class on top of the outdated Riemann integral.
4/23/2026, 11:45:53 AM