How to solve an n-dimensional integral?

[RoyGBiv12]

Homework Statement

http://imageshack.us/photo/my-images/404/1322740516223.png/

(http://imageshack.us/photo/my-images/404/1322740516223.png/)

Homework Equations

see image

The Attempt at a Solution

umm, I have no idea how to begin case n=2

 

[jackmell]

How about start with a plot? Then get rid of that ugly looking integral, then just focus entirely on n=2 like that's all you got to do.

$$F_2(t)=\int_0^t\int_0^t f(\text{min}(x,y))dydx=\mathop\iint\limits_{\text{blue}} f(x)dydx+\mathop\iint\limits_{\text{red}} f(y)dydx$$

or is it the other way around? Need to check.

 

[RoyGBiv12]

Hmmm, so you split it into two integrals to cover both cases (f(x) or f(y) being the min value). Are all four bounds still 0 to t? And how should I go about finding the antiderivative of that expression as was done in case n=1. Does this require Green's theorem?

 

[jackmell]

Hmmm, so you split it into two integrals to cover both cases (f(x) or f(y) being the min value). Are all four bounds still 0 to t? And how should I go about finding the antiderivative of that expression as was done in case n=1. Does this require Green's theorem?

What do you think? Come up with something. Show some work. That's a requirement in this sub-forum. Then try and post what you think are the limits using Latex. That the language we use in here to make nice math symbols. See:

https://www.physicsforums.com/showthread.php?t=546968

or just post them as text just to show you're trying. Or you can do a quote on my post to see how I coded the latex for that expression above. And I tell you what, suppose I have an integral function:

$$f(t)=\int_a^t g(x,t)dx$$

how would I find f'(t)? Leibniz right?