Double integral problem, conceptual help.

[RJLiberator]

Homework Statement

Find the volume: Prism formed by x+z=1, x-z=1, y=2, y=-2, and the yz-plane.

Homework Equations

The Attempt at a Solution

Okay, so I sketched the drawing and I found that I could take the upper region of the xy-plane with respects to x and z and a triangle was formed.
The bounds were from 0 to 1 with respects to x and from 0 to z=1-x with respects to z.
So i integrated dzdx and got the answer of 1/2 for the triangle.

I then multiplied the triangle area by 2, since we have to take the volume of above the xy-plane and below the xy-plane which due to symmetry, I assume this works.

I then multiplied by 4 as y=-2 and y=2 distance results in 4 units.

So my answer came to be "4".

This seems a bit too...easy, however, I can't find anything conceptually wrong with my procedure.

Did I perform this operation correct?

 

[fourier jr]

If y = f(x, z) ≡ 2 & by only calculating the part of the prism in one octant then multiplying by 4 (symmetry) I got ##4\int_{0}^{1}\int_{0}^{1-x} f(x, z) dz\,dx = 4\int_{0}^{1}\int_{0}^{1-x} 2\,dz\,dx = 4##. That's the same as I got by just doing (length)x(width)x(height)/2 so it seems right...

 

[RJLiberator]

Oh wow, that is awesome. It made sense to me in my head, but it seemed like I was missing something. I'm glad we got the same answer. Thank you for checking.