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I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about integration in engineering are pratically pure physics. So, as I said, I'm graduating, but I'm from Brazil, education here is the same thing as nothing (belive me, it really sucks, I see people come out of physics's college without knowing who Maxwell was..), and I read my calculus books and see they all use Riemann integrals (of course, they don't say that..); but, in my searchs, I see a lot of Lebesgue integration, especially concerning problems of calculating the center of mass of an object, in continuum mechanics, etc. So, here's my question:

  1. Lebesgue integration isn't the most easy thing in the world, I don't have a lot time, and education here sucks, should I spent my time studying Lebesgue integration instead of Riemann's?

  2. Which one do you guys use more in physics with applications in engineering?; of course, the last one it's a little trouble, because engineering just "borrow" from physics. But, in general, which compensates more?

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    $\begingroup$ There's a good Richard Hamming quote that I think is quote appropriate for this situation: "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." $\endgroup$ – DumpsterDoofus Apr 22 '14 at 21:35
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    $\begingroup$ Thank you. @Bombyx, I think about math SE, but since I will only use integration applied to physics, I thought you guys would be the best to answer.. $\endgroup$ – Ricardo Apr 22 '14 at 21:37
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    $\begingroup$ @user5462: In short, I probably wouldn't waste too much time on Lebesgue integration, unless it's really absolutely necessary. Instead, try to focus on what's practical and useful for the work that you are studying for, and don't worry too much about slippery mathematical details like Lebesgue integration. Most engineers I know have never even heard of Lebesgue integrals, and it was never covered in any of my physics classes either. $\endgroup$ – DumpsterDoofus Apr 22 '14 at 21:39
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    $\begingroup$ @user5462: I would encourage you to learn it any way, it is abstract but is useful for many purposes. Once you know real analysis and functional analysis well, you are prepared to study more advanced mathematical physics, which often involve a fair amount of PDE. $\endgroup$ – Bombyx mori Apr 22 '14 at 21:42
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    $\begingroup$ ... by Lebesgue himself of his idea to his friend Paul Montel. Ponder this and study carefully how the Lebesgue integral takles the integral of the function $\phi:[0,1]\to\{0,\,1\}$ where $\phi(x)=1$ if $x\in\mathbb{Q}$ ($x$ is rational) and $\phi(x)=0$ if $x$ is irrational ($\int\,\phi\,dx=0$, BTW) and once you've done that, you'll have a good grasp of the gist of the idea and you may be in a better position to know whether you need to study it further. $\endgroup$ – Selene Routley Apr 23 '14 at 6:20