Volume of cylinder bounded by two dependent planes, ideas?

[dude899]

Homework Statement

[/B]
Calculate the volume bounded by the plane/cylinder x^2+y^2=1 and the planes x+z=1 and y-z=-1.

Homework Equations

/ The attempt at a solution[/B]

It is pretty basic triple integral in cylindrical coordinates. For some reason, I can't get the right answer. I'm using bounds: 1-x ≤ z ≤ y + 1, -3π/4 ≤ θ ≤ π/4, 0 ≤ r ≤ 1.

The shape is a cylindrical wedge of sorts but it is so twisted I'm not sure this is correct. Any ideas?

 

[LCKurtz]

Homework Statement

[/B]
Calculate the volume bounded by the plane/cylinder x^2+y^2=1 and the planes x+z=1 and y-z=-1.

Homework Equations

/ The attempt at a solution[/B]

It is pretty basic triple integral in cylindrical coordinates. For some reason, I can't get the right answer. I'm using bounds: 1-x ≤ z ≤ y + 1, -3π/4 ≤ θ ≤ π/4, 0 ≤ r ≤ 1.

The shape is a cylindrical wedge of sorts but it is so twisted I'm not sure this is correct. Any ideas?

This is a tricky problem. The problem is that the two planes cross each other inside the cylinder. If you set the z values equal you get $y = -x$. If you look at the projection in the $xy$ plane that line divides the circle into two regions, $$-\frac \pi 4 \le \theta \le \frac {3\pi} 4 \text{ and }\frac {3\pi} 4 \le \theta \le \frac{7\pi} 4$$I think you will find that in the first region $z=1+y$ is the upper surface and in the second region $z=1-x$ is the upper surface. Work out the integrals for those two cases and see if that helps.

Edit, added: Here's a picture. It isn't oriented in the usual direction in order to get a nice view of it. But you can see the shape of it:

 

[dude899]

Ok, so I get that the volume would then be V_tot = V₁ + V₂ = (2 * √(2)) / 3 + (2 * √(2)) / 3 = (4 * √(2)) / 3

Is this correct, does someone get different answers?

 

[LCKurtz]

Ok, so I get that the volume would then be V_tot = V₁ + V₂ = (2 * √(2)) / 3 + (2 * √(2)) / 3 = (4 * √(2)) / 3

Is this correct, does someone get different answers?

Not quite what I got. Let's see your integrals.
[Edit] Woops! Cancel that. I agree with your answers.