What Shape Does This Inequality Represent in R3?

[fk378]

Homework Statement

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

Homework Equations

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

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?

 

[Dick]

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]

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.

 

[fk378]

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]

y can be anything and the inequality is still satisfied.