# Describe this region of R3

1. Homework Statement
Describe in words the region of R3 represented by the inequality x^2 + z^2 <= 9

2. Homework Equations
Equation of a sphere= (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

3. The Attempt at a Solution
Since there is no y value in the given inequality, I stated that it would be points in or on a circle on the xz-plane with center at the origin, and the radius is 3 with respect to the xy-plane.

However, my book says this inequality describes a cylinder of radius 3 with y-axis. Can someone explain this to me please? How can it be a cylinder? And why is the radius with the y-axis and not with the xy-plane?

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Dick
Homework Helper
The intersection of the set with the x-z plane is a disk, right? Since y does not appear in the inequality, y can be anything as long as the x-z coordinates are in the disk. This is the same as saying the it's the union of all lines passing through the x-z disk parallel to the y axis. Isn't that an infinite cylinder?

tiny-tim
Homework Helper
However, my book says this inequality describes a cylinder of radius 3 with y-axis.

And why is the radius with the y-axis and not with the xy-plane?
HI fk378!

You're right … the radius is not with the y-axis. But … you're misreading the book. The cylinder has radius 3, and the axis of the cylinder (the infinite line which runs exactly through the centre of the cylinder) is the y-axis. The intersection of the set with the x-z plane is a disk, right? Since y does not appear in the inequality, y can be anything as long as the x-z coordinates are in the disk. This is the same as saying the it's the union of all lines passing through the x-z disk parallel to the y axis. Isn't that an infinite cylinder?
So if a variable is not given bounds in the inequality then it means that it can take on any value? It doesn't have to be y=0 always?

Dick