How to Set Up a Triple Integral Over a Tetrahedral Region?

[DrunkApple]

1. The problem statement, all variables and given known data
f(x,y,z,) = e^{2x-z}
W: x + y + z ≤ 1
x, y, z ≥ 0

Homework Equations

The Attempt at a Solution

For each domain, could you check it please?
This is the only triple integral that's haunting me

0 ≤ x ≤ 1-y-z
0 ≤ y ≤ 1-x-z
0 ≤ z ≤ 1-z-y

 

[Mark44]

1. The problem statement, all variables and given known data
f(x,y,z,) = e^{2x-z}
W: x + y + z ≤ 1
x, y, z ≥ 0

Homework Equations

The Attempt at a Solution

For each domain, could you check it please?
This is the only triple integral that's haunting me

0 ≤ x ≤ 1-y-z
0 ≤ y ≤ 1-x-z
0 ≤ z ≤ 1-z-y

I don't think you're on the right track here, at all. If you haven't already done so, sketch the region over which integration is to be done. The region is a tetrahedron that is bounded by the three coordinate planes and the plane x + y + z = 1. Where this plane intersects the x-y plane will play a role in your description of the region.