# Is this correct sytntax: the slice is a semicircle **parallel** to the x axis when perpendicular to y with triangle base along the x and y axes?

Find the volume where the base is a triangle with the vertices: (0,0),(4,0),& (0,4). The cross section is a semicircle and is perpendicular to the y-axis and is parallel to the x-axis.

I get everything up to the "parallel to the x-axis" part. Okay, so we are not using the z-axis for this course but isn't the cross-section parallel to the z-axis?

Okay, so I guess the professor was talking about was the base of the cross-section, not the cross-section.

I get that.

However, I have looked around the internet and in my multiple text books and I have never seen this type of problem explained as "parallel to the x-axis" so I want to know if this is incorrect syntax.

If the base is along the x and the y axis, how is the cross-section itself parallel to x?

• The wording is a bit awkward. The best interpretation (as you seem to suggest) is that "perpendicular to the $y$-axis" describes the plane of the semicircular cross section, while "parallel to the $x$-axis" describes the diameter of the semicircle itself. (For a plane, "perpendicular to $y$" already guarantees "parallel to $x$ (and $z$)", so using both to describe the cross-sectional region would be redundant.) I might've phrased things like this: "Cross sections, in planes perpendicular to the $y$-axis, are semi-circles (above the $xy$-plane) with diameters spanning the figure's base."
– Blue
Commented Oct 15, 2017 at 4:04
• @Blue You see, this is what confused me. I would have understood that. Commented Oct 15, 2017 at 4:15
• @Blue I probably spelled that wrong. My spell check is British English, not U.S. Commented Oct 15, 2017 at 4:16

A triangle has vertices: O (0,0),X(4,0),& Y (0,4). Find half cone volume enclosed when triangle $YOX$ (slant line and base radius) are rotated $180^0$ about y-axis.