Volume of Bounded Region: Solving Limits for Multiple Integration

[windy906]

I'm having problems finding the limits for the follwing integration, can anyone help?

Find the volume bounded by x+2z=4 and 2y+z=2 with x,y and z>=0

 

[windy906]

I think the following limits apply but can't find the upper one for z, or in which order they should go.

x: 0 -> 4-2z
y: 0 -> 1-0.5z
z: 0 -> ?

 

[Office_Shredder]

It seems like if you pick any z>=0, you can find x,y that satisfy the boundary condiitions. Hence, z should be between 0 and $$\infty$$

 

[D H]

It seems like if you pick any z>=0, you can find x,y that satisfy the boundary condiitions. Hence, z should be between 0 and $$\infty$$

There are five boundary conditions:

1. x+2z<=4
2. 2y+z<=2
3. x>=0
4. y>=0
5. z>=0

Picking a value for z beyond some finite limit will not work.

 

[windy906]

So z <= 2!

Thank you so much.