3 Integrals You Won't See in Calculus (And the 2 You Will)

In Calculus, we usually learn the Riemann integral, or sometimes the Darboux integral in disguise. But there are many problems these integrals can't solve! Like if we want to integrate a function which is discontinuous everywhere, or if we want to integrate with respect to a random process. Let's explore 5 different integrals, starting with the 2 you might see in Calculus, and then 3 more advanced integrals that are often only seen in graduate school -- the Riemann-Stieltjes integral, Lebesgue integral, and Itô integral!

00:00 Introduction
00:32 Level 1 -- Riemann Integral
01:58 Level 2 -- Darboux Integral
04:00 Level 3 -- Riemann-Stieltjes
07:02 Level 4 -- Lebesgue Integral
09:57 Level 5 -- Itô Integral

[mtaur]

Darboux is usually lumped in as a case of Riemann, which is a somewhat generalized concept after all. If you know f is continuous, then you can just take left endpoints and equal width intervals, as far as the limit is concerned, but it has to be proved that the choices don't matter.

As far as arbitrary partitions go, insisting on equal spacing could make it annoying to prove that you can split up integrals from 1 to e and then e to pi, for example. By the time you justify the validity of uniform steps, you have already considered arbitrary partitions as the max width goes to zero, and may as well have allowed them from the start.

It's a lot of fine print but if f is continuous, uniform intervals and left endpoints go brrrr.

[xashans]

On sets of measure zero, always bet on Lebesgue

[harriehausenman]

Brilliant video!👍 Exactly the right amount of depth for me 🤗
Thanks for putting effort into the production and having great audio and video, and double thanks for not using negatively biased graphics. 🙏

[SeanRaleigh]

I didn't know about the Itô integral, so I learned something new today. Nice!

[fdileo]

How to explain integral calculus in 12 minutes... You nailed it perfectly!

[jamiepianist]

I'm so interested in Itô integrals now, Dr. Sean is the GOAT

[jesuseduardobanosgonzalez]

Woah, I am astounded by how easy to understand you made the concepts of the more complicated integrals!

[kindreddarkness]

You are a fantastic teacher. I am new to calculus, and have really wondered why we would need more than a "general integral." This video not only answered the question, but also justified what the hell derivatives are actually measuring (and why we bother taking them). Thank you.

[awesomesam]

Great video, but at 9:55 it's more correct to say that we get an irrational number *almost* every time (i.e. with probability one, but it's still technically possible to get a rational number).

[alexbstl]

Re: peeking into the future, that’s not entirely true. The Stratonovich Integral (midpoint rule) is also an adapted process (meaning it can’t see the future) and even results in the standard chain rule when taken in differential form. The real reason people use an Ito integral is that it is a martingale. Admittedly, this is a bit technical.

[harjooni]

If I'm not mistaken, by "random process" you mean a stochastic process, i.e. a random variable with an index number (often, though not always, interpreted as time) attached?

In the case you showed, the index number has an interpretable direction so talking about it as time is meaningful and some authors would call the process causal.

But what happens if we give up on the causality assumption (as some do in time series analysis, though that is in discrete time) and let the "future" (i.e. events with a high index number) affect "today"? Is the Ito integral still valid?

[ennyiszizlak]

Hi Dr Sean! The video was very insighftul and easy to comprehend. Thank you very much for your work. I am looking forward to lean more maths in an MBA program than in my undergrad in finance. I was wondering about what measury theory is and how brownian is relevant to stohastic analysis. Thanks again, looking forward for more videos in the future :)

[Cpt.Zenobia]

Love the visualizations always makes it easier.

[jdp]

Thank you. Very nice to see this info being simply explained. In the Ito integral, it's interesting how 2nd order terms are important (in a standard deviation sense) because of the nature of the random process, whereas in the other types of integration presented the 2nd order terms are considered insignificant (zero). Would have been nice (a luxury) to include the Generalized Riemann Integral (uses a different type of partitioning).

[redknight]

awesome video thank you very much for showing this is a manner so simple yet so complete!

[taranmellacheruvu]

I’d just like to say on thing regarding Itô integrals: one of the reasons it’s mathematically tricky to work with is that you’re integrating wrt a fractal function (Brownian motion is a fractal) meaning that you can’t really make your partitions infinitely thin in the same way as Riemann integrals. This results in some interesting rules regarding differentials

[stochasticxalid]

Wonderful, Dr Sean. A new subscriber here.

[ethanfletcher]

Expectation values in quantum mechanics makes so much more sense now

[somezw]

Excellent video, but I would like to point out that at 6:42 the distribution function should be right-continuous as it is a property of all cdfs

[VeteranVandal]

Incredibly enough, I knew all of them. I need to learn #4 for real tho.