Take the function X^1/X, and plot it.
Take the function X^X, and plot that.
What is the area between the two functions, from 0 to 1?
I’m guessing some value related to e, but I never passed calculus.
(derivatives…first, second, third integrals, inflection points:confused: 
)
The area is most likely infinite, as x[sup]1/x[/sup] goes to infinity as x goes to zero (from the right, which is all that matters here).
hmmm…the problem with this is, X^(1/X) isn’t well-behaved close to zero (as X → 0, 1/X → infinity, so X^(1/X) → infinity). You could measure the area under the curve X^(1/X) from, say, .0000001 to 1, but you can’t get infinitely close to 0 because the closer you get, the bigger the area gets, until you hit 0 and the function stops being well-defined.
That and, I don’t have the foggiest damn idea of how to integrate X^X dx. Or X^(1/X) dx.
Why do you need to know this anyway? 
I plotted it in Excel, and it looks to me like x^(1/x) goes to zero as x goes to zero.
If it does go to infinity, how would you integrate those functions to find the area from, say, 1/2 to 1.
I don’t think so, ultrafilter. As a f’rinstance, if x = 1E-n
1E-n[sup]1/1E-n[/sup] = 1E-n[sup]1E+n[/sup]
n = 1, x[sup]1/x[/sup] = 0.1[sup]10[/sup] = 1E-10
n = 2, x[sup]1/x[/sup] = 0.01[sup]100[/sup] = 1E-200
n = 3, x[sup]1/x[/sup] = 0.001[sup]1000[/sup] = 1E-3000
Maybe. It’s still a bitch to integrate, though–you may just have to use numerical means for this one.
Incidentally, is there a name for the x^x class of functions? Exponentials? Polynomials? Exponential polynomials?
-b
Doh! (a number < 1)^(a big honkin’ number) != (another big honkin’ number)
x[sup]1/x[/sup] does indeed approach zero, not infinity. I’m still not sure how to integrate it though. If there is a way, then here’s what you do:
Take the integral from 0 to 1 of x[sup]x[/sup], and subtract the integral from 0 to 1 of x[sup]1/x[/sup]. This works because x[sup]x[/sup] > x[sup]1/x[/sup] for 0 < x < 1 .
I say, just take a bunch of values, put it in Excel, and do it by hand.
around 0.43
I get about 0.435
Doing the - take a bunch of values, throw them in excel, and guess and round, I’m getting ~.43… but that makes a number of wild assumptions, among which is that I remember how to do this at all. (It’s been a while.)
You guys did it right.
using 0.001 increments, I get 0.43043
FWIW, Mathematica (or at least the version on integrals.com) doesn’t know how to handle those functions. I take that to mean that no one knows.
Doing it numerically in Mathematica 4.2 I got 0.429934
My calculator’s usually pretty good about this, and it gets 0.42993371014878. I wouldn’t count on all those decimals being correct, but probably most of them.
It’s still possible that this problem can be solved analytically, even if the indefinite integral of x[sup]x[/sup] cannot. I’m looking for a trick, but no luck yet…