the region given is
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
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.
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
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
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
so my integral would look something like
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.