# Triple integral forming a solid region

1. Sep 15, 2016

### nysnacc

1. The problem statement, all variables and given/known data

2. Relevant equations
Fubini's theorum

3. The attempt at a solution
I drawn the diagram with the limits (for x, y, and z)
and come up with something with 4 faces, 5 corners, 8 edges

is that something you guys got? Thanks

2. Sep 15, 2016

### andrewkirk

The number of corners and edges are correct. I don't think the number of faces is.

3. Sep 15, 2016

### BvU

I also come up with a different number of faces

4. Sep 15, 2016

### nysnacc

Like how many as what am I doing wrong??

The x limit is 0 and z, so for z, it is between y and 1, so I put 1 as upper limit for x.

5. Sep 15, 2016

### nysnacc

6. Sep 15, 2016

### LCKurtz

One of the faces is part of the plane $x=z$. You don't have that in your picture.

7. Sep 15, 2016

### BvU

I count five faces on your drawings too...
z = 1 is one plane, a square face
x = 0 is a triangular face
y = 0 is a triangular face, not a square face: x runs from 0 to z, not from 0 to 1.
the other two triangular faces have x=y=z as edge in common

8. Sep 15, 2016

### nysnacc

so it's like a square on top with the edges pointering to a single point on bottom?

9. Sep 15, 2016

### andrewkirk

That's right - like the Great Pyramid (disregarding the references to 'top' and 'bottom'). This shape is known as a 'square pyramid'. The word 'square' is included to distinguish it from a tetrahedron or triangular pyramid. But in common language, the word 'pyramid' usually implies a square base.

10. Sep 15, 2016

### LCKurtz

Now that you have it, here's a nice picture:

Now the trick for you is to set up the six integrals. Do you see why when you do $dz$ first something is a bit more complicated?