Volumes with triple integrals, aka I suck at geometry

[Gauss M.D.]

Homework Statement

Calculate the volume of the body that is bounded by the planes:

x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2

Homework Equations

The Attempt at a Solution

I made a variable substitution

u = y+z
v = y-z
w = x

which gave me the new boundaries

u+w = 2
v+w = 0
v = 0
u = 0

Problem is, I must have slept late the day they taught this is class. What do I do to determine which way these inequalities go?!?

 

[Mandelbroth]

Homework Statement

Calculate the volume of the body that is bounded by the planes:

x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2

Do you think you need calculus?

$$z=x+y \\ z= -x -y +2 \\ z=y \\ z=-y$$

That should be pretty simple...

 

[verty]

You should not change variables before you have set up the integral. Do that first.

 

[haruspex]

To get an idea of the shape, try to find the vertices.