Recent content by The_eToThe2iPi

But doesn't this also not have a Riemann integral over that same domain since ##\int^{1}_{\delta}\frac{1}{x}dx=[ln(x)]^{1}_{\delta}=ln(\frac{1}{\delta})## but ##ln(\frac{1}{\delta}) \to \infty## as ##\delta \to 0##?

What domain are you integrating ##f(x)=\frac{1}{x}## over? Does an improper Riemann integral exist for ##f(x)=\frac{1}{x}## over that domain? The reason I used ##sin(x)/x## was because it is an alternating function and so needs to be broken down into it's positive and negative parts in order to...

To be honest, I feel like these are the sort of questions that require me to read a book (and I'll probably have even more afterwards). Thank you all so much for answering my questions and pointing me in the right direction.

Wow, just imagine if students had to buy all their own textbooks! £152! Fortunately I think the university has a copy of that one so I won't have to sell my kidney :) The page on gauge integration mentions that this integral can be extended more general domains and I was wondering whether...

Ahh, that's interesting. I didn't think about that. But I still feel that there is a difference between the two instances. In your case we are evaluating two separate Riemann integrals, finding they are infinite, and then asking what their difference is like asking what is ##(a+\infty)...

So I'm studying a course on measure theory and we've learned that the Lebesgue integral of a real function is (loosely) defined as the total area over the x-axis minus the total area under the x-axis. This seems to me to be limited because these areas can both be infinite but their difference...