A Little Bit of Math
February 7, 2005 – 11:30 pmDo you know what makes a function “integrable”? I thought I did, but as usual, I really didn’t, at least until today. Two buzz words kept floating to mind: “continuous” and “bounded”. Of course neither of those matter. So, what makes a function “integrable”? A function f(x) is integrable on some domain (a,b) if the integral of f(x) on this domain exists (i.e., is finite). That’s it.
I learn something new every day, even if it is something I was supposed to know when I was in the seventh grade. Oh well.
3 Responses to “A Little Bit of Math”
Just in case it makes you feel better, the sort of integration that one might learn in the seventh grade is Riemannian integration (summing lots of increasingly narrow rectangles). This type of integration DOES fail for some functions that are not continuous, for example integrating f(x) on x=[0,1], where f(x)=1 for x rational, otherwise f(x)=0. The integral of this is zero, using Lebesgue integration, but is not defined using Riemannian integration.
And yes, I get to think about this sort of thing every day. Lucky me!!!
By paul.za on Feb 9, 2005 at 11:51 pm
I think you put too much stock in those buzz words, since I know for a fact you did not think a function had to be continuous to be integrable. Everyone’s favorite example: the step function.
Bounded is trickier, but Integral[Log[x], {x, 0, 1}] is a nice example. Along the same lines, my favorite mathematical trick is the following:
Take f(x)=1/x on the interval [1, inf) and rotate it about the x-axis. You get a horn like figure. I leave it as an excersize to the reader to show that, while the horn holds a finite volume, it has infinite surface area. Stated more thought provokingly: it can hold a finite amount of paint but requires an infinite amount of paint to paint it.
My highschool math text called it “Gabriel’s Horn”.
By MDA on Feb 10, 2005 at 1:49 am
Well, of course I put too much stock in those buzz words, because when I was getting grilled about it, I had to gravitate toward something, and those two seemed perfectly reasonable at first glance. But, you’re right, within ten seconds I realized they were both bogus, but I thought they were a good starting point. At the time, I had just thought a function being integrable meant something (okay, okay…it means _something_), but it turns out the vocabulary is (at least seemingly) tautological.
By jjk on Feb 10, 2005 at 10:59 am