Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to figure out how to switch the order of integration for this problem. I'm given the region (I don't know how to use latex, maybe someone can clean this up for me?)

$z$ from $0$ to $x+y$

$y$ from $0$ to $1-x$

$x$ from $0$ to $1$

That was for the integral $dz\;dy\;dx$ and I have to change it to $dy\;dx\;dz$. I think maybe I drew out the region wrong and that's why I can't figure this out. The one I drew looks like a right triangle. In the first octant The $x$ axis from 0 to 1 and the hypotenuse of the triangle is the line $1-x$ then it goes out in $y$ from 0 to 1 (I can upload the picture if it helps). I don't think I'm taking the $x+y$ part into account anywhere? The new integral I came up with is

$y$ from $0$ to $1$

$x$ from $0$ to $1-y$

$z$ from $0$ to $1$

share|improve this question
2  
I don't think that $x$ should depend on $y$, if you go in this order. –  Dylan Moreland Jan 21 '12 at 19:26
add comment

1 Answer

Write your limits of integration as inequalities:

$$0\le z \le x+y$$ $$0\le y \le 1-x$$ $$0 \le x \le 1$$

and you should be able to see $0 \le y \le 1-x \le 1$ and $0 \le z \le x+y \le x+1-x =1$. You also have $z-x \le y \le 1-x$ so you could have

$$\max(0, z-x) \le y \le 1-x$$ $$0 \le x \le 1$$ $$0\le z \le 1$$

making your formulation

$$\int_{x=0}^1 \int_{y=0}^{1-x} \int_{z=0}^{x+y} dz \, dy \, dx = \int_{z=0}^{1} \int_{x=0}^1 \int_{y=\max(z-x,0)}^{1-x} dy \, dx \, dz $$

but it might be simpler if you split this depending on whether $x$ or $z$ is bigger so

$$z-x \le y \le 1-x$$ $$0 \le x \le z$$ $$0\le z \le 1$$

and

$$0 \le y \le 1-x$$ $$z \le x \le 1$$ $$0\le z \le 1$$

making your formulation

$$\int_{x=0}^1 \int_{y=0}^{1-x} \int_{z=0}^{x+y} dz \, dy \, dx = \int_{z=0}^{1} \int_{x=0}^z \int_{y=z-x}^{1-x} dy \, dx \, dz + \int_{z=0}^{1} \int_{x=z}^1 \int_{y=0}^{1-x} dy \, dx \, dz $$

share|improve this answer
    
Thanks this helps but I don't get the second set of inequalities at all. If x is less than 1 then 1-x should be greater than zero, right? I see your answer works but can you explain the inequities a bit more? –  Needsmathhelp Jan 21 '12 at 21:47
    
If you are asking about $\max(0, z-x) \le y$, you know $0\le y$ and you know $z \le x+y$ which is the same as $z-x \le y$. So $y$ is greater than or equal to both $0$ and $z-x$ and so greater than or equal to the maximum of those two. Note that $z$ can be less than, equal to or greater than $x$, and it is sensible to treat the less than and greater cases differently, in which case the $\max$ can be removed. –  Henry Jan 21 '12 at 22:36
    
I got it now, thanks! –  Needsmathhelp Jan 22 '12 at 0:23
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.