Riemann vs Lebesgue Integral

In this video, I show how to calculate the integral of x^3 from 0 to 1 but using the Lebesgue integral instead of the Riemann integral. My hope is to show you that they indeed produce the same answer, and that in fact Riemann integrable functions are also Lebesgue integrable. Enjoy!

[danielberkowitz]

I'm a physics Ph.D student and I must say you are a gift doctor. We need more people to enthusiastically teach advanced math in such a clear and concise way with a smile. You are doing an outstanding service, keep the excellent work.

[nullplan]

Important caveat: The Lebesgue integral is only an extension of the normal Riemann integral, not of the improper Riemann integral. The Dirichlet integral, for example, (Int(0..inf, (sin x)/x dx)) is improperly Riemann integrable, but not Lebesgue integrable. At least, that's what Wikipedia claims.

[ivanglass]

You explain everything with huge smile and a lot of passion i really enjoyed watching this !!

[cameronspalding]

Riemann integral: partition [a, b] where a=x0<x1<x2<...<xn=b
Let x i-1 < ei < x i
Riemann sum = sum (i=1 to n) f(e i)*(x i - x i-1) using partitions with smaller meshes gives a more accurate result
Lebesgue integral: approximate the function with indicator functions f(x)= sum a i * indicator (set X i)
Lesbegue integral = sum for i a i * m(X i)

[antoniss]

Good job man!You have a good personality and an interesting presentation way.

[sebastianramirezcaseres]

Greetings from Colombia doc.
Thanks so much for the explanations

[jensmalmquist]

What a matchup. Truly the Logan Paul vs KSI of its day.

[saadslaoui]

Thank you. You just made me more educated ...

[klausgiesselmann]

I learnt that with the Lebesgue integral the area under the curve is sliced horizontally.
Or did I miss something?

[hichamallam]

I LIKE 5:45 see the middle triangle smiling 😃 😊 with close eyes ( i-1)/N

[justcarcrazy]

Yeah, I'm not any less confused...

[riccardopratesi]

Shouldn't the width written at 5'10" be 1/N, instead of i/N?

[MrRyanroberson]

I wonder.... What is the actual length of the line between f(a) and f(b)? Like for f(x)=sqrt(1-x) the length from -1 to 1 is π, so what about x^3 and others?

[tommylofgren]

This is such a great video! Thank you!

[bettkitty]

I thought Riemann was vertical rectangles and Lebesgue was horizontal rectangles (based off pictures on Wikipedia article).

[MiroslavMakaveli]

We need more push on Analyse II :P Can you show us Taylor's Theorem?
Can you do some max min values of two or more variables with graphs or something like will be nice to see.

[duckymomo]

What’s an example of neither integrable?

[adityamanimishra]

nice explanation, interesting presentation,

[saitaro]

This looks like a special case, since we know the formula for the sum of the cubes of the integers from 1 to N, but what about the global case, does it exist for Lebesgue integral, like *F(b) – F(a)* for Riemann integral? How to integrate cos(x) with Lebesgue, for example?

Thanks for the video btw :)

[karstenmeinders]

So I understand the Lebesgue integral is a true extension when the Riemann integral does not work, but for "simple" functions like polynomials the resulting funtioncs will be same same. What about trig, exp, ln functions and their combinations? Do we need the chain rule, U-substition and the like as well? Thank you very much! And continue your great work!