Calculating Volume of a Cylindrical Wedge

[unscientific]

Homework Statement

The problem is attached in the picture.

The Attempt at a Solution

What bothers me is that they say the wedge is bounded by x + z = a. Doesn't this imply that the calculated volume should only be half of what is written in the answers? I'm aware that the plane x + z = a can refer to either the left (negative) side or the right (positive) side. Do we choose one or simply take both?

Limits for x: 0 to a
Limits for y: 0 to √4ax
Limits for z: 0 to a-x


However, in http://mathworld.wolfram.com/CylindricalWedge.html the wedge is given as the full volume.

 

[clamtrox]

You have two possibilities: if a>0, then x>0 and z>0. If a<0, then also x<0 and z<0. No value of a works for both regions.

 

[LCKurtz]

Homework Statement

The problem is attached in the picture.

The Attempt at a Solution

What bothers me is that they say the wedge is bounded by x + z = a. Doesn't this imply that the calculated volume should only be half of what is written in the answers? I'm aware that the plane x + z = a can refer to either the left (negative) side or the right (positive) side. Do we choose one or simply take both?

Limits for x: 0 to a
Limits for y: 0 to √4ax
Limits for z: 0 to a-x

Assuming $a>0$, the slanted plane is the "roof" of the solid. I see no reason to limit $y$ to only being positive.

 

[unscientific]

Assuming $a>0$, the slanted plane is the "roof" of the solid. I see no reason to limit $y$ to only being positive.

I see! I initially thought that the x + z = a plane refers to the vertical plane i.e. the x-z plane. But now that you mention it, it actually refers to the plane that is sloping downwards from z = a to x = a, like a ramp.