1. ## describe given region

the region given is

$x=0;y=0;z=0;x+y=4;x=z-y-1$

I plotted it on grapher

so I get a region with five sides, the problems i have encountered so far I was able to slice with a z, x or y plane and get a 2-D region that I slice with a vertical or horizontal line then find the change of the other variable. then I could get all three limits of integration.

I can't decide how I should slice this. I though about with the z=0 plane but then I on 2-D image but theres another view that gives a different 2-D image that needs to be described. seems like I'll have that issue no matter how I slice

2. ## Re: describe given region

if I slice with the x=0 plane or y = 0 plane then I get similar 2-D images but that still leaves out other important information

3. ## Re: describe given region

I presume that by "region given" you mean the region bounded by those planes. Of course, x= 0, y= 0, and z= 0 are the coordinate planes so this region is in the first octant. x+ y= 4 is the plane crossing x= 0 in the line y= 4 and crossing the plane y= 0 in the line x= 0.

x= z- y- 1, equivalently, x+ y- z= -1, cuts the x= 0 plane in the line y- z= -1, the y= 0 plane in the line x- z= -1, and the z= 0 plane in the line x+ y= -1. The two planes, x+ y= 4, equivalent to x= 4- y, and x= z- y- 1 in the line x= 4- t, y= t, z= 5.

4. ## Re: describe given region

yes bounded by, I used the wording from the question

5. ## Re: describe given region

Originally Posted by HallsofIvy
I presume that by "region given" you mean the region bounded by those planes. Of course, x= 0, y= 0, and z= 0 are the coordinate planes so this region is in the first octant. x+ y= 4 is the plane crossing x= 0 in the line y= 4 and crossing the plane y= 0 in the line x= 0.

x= z- y- 1, equivalently, x+ y- z= -1, cuts the x= 0 plane in the line y- z= -1, the y= 0 plane in the line x- z= -1, and the z= 0 plane in the line x+ y= -1. The two planes, x+ y= 4, equivalent to x= 4- y, and x= z- y- 1 in the line x= 4- t, y= t, z= 5.
I'm not following you.

"x+ y= 4 is the plane crossing x= 0 in the line y= 4 and crossing the plane y= 0 in the line x= 0." ??

the plane x+y=4 crosses the y=0 plane at y=4 and crosses the x=0 plane at x= 4

I can see that the bounded region is in the first octant. the question is looking for the intervals of integration for x,y,z. I get a surface with 5 faces. all my other problems up to this point I was able to slice with a plane which then allowed me to have a 2-space coordinate system so that I could find the intervals. In this I don't know where to slice and it seems I would need to slice it a couple or more ways to get everything or I am over complicating it.

6. ## Re: describe given region

I believe I figured it out. the question just wants the limits of each variable in interval form

so if I fix x based on the restrictions I can see from x=0 and x+y=4 I get $0 \le x \le 4$

now I "slice" with the x=0 plane to get a 2-space coordinate plane of z vertically and y horizontally

I know y has a "left" boundary by the y=0 plane and from x+y=4 I know the "right" boundary on y is 4

so know I have $0 \le y \le 4$

now for z it has a lower boundary from the z=0 plane and if I "slice" this coordinate plane with a fixed y I can see how z changes as y goes from "left" to "right"

and now solving the system created by x+y-z-1=0 and x+y=4 to get the intersection then solving for the equation of the line created by x+y-z-1=0 from y=0 to x+y=4

I get z bounds

$0 \le z \le \frac{1}{2}y+1$

so my integral would look something like

$\int_{0}^{4}\int_{0}^{4}\int_{0}^{\frac{1}{2}y+1} f(x,y,z)dzdydx$

7. ## Re: describe given region

Originally Posted by Jonroberts74
I'm not following you.

"x+ y= 4 is the plane crossing x= 0 in the line y= 4 and crossing the plane y= 0 in the line x= 0." ??
That was a typo. I meant, of course, "x= 4". Sorry.

the plane x+y=4 crosses the y=0 plane at y=4 and crosses the x=0 plane at x= 4

I can see that the bounded region is in the first octant. the question is looking for the intervals of integration for x,y,z. I get a surface with 5 faces. all my other problems up to this point I was able to slice with a plane which then allowed me to have a 2-space coordinate system so that I could find the intervals. In this I don't know where to slice and it seems I would need to slice it a couple or more ways to get everything or I am over complicating it.