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Homework Help: Describe this region of R3

  1. Mar 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Describe in words the region of R3 represented by the inequality x^2 + z^2 <= 9

    2. Relevant 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?
    Last edited: Mar 29, 2008
  2. jcsd
  3. Mar 29, 2008 #2


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    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?
  4. Mar 29, 2008 #3


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    HI fk378!

    You're right … the radius is not with the y-axis. :smile:

    But … you're misreading the book. :frown:

    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. :smile:
  5. Mar 29, 2008 #4
    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?
  6. Mar 29, 2008 #5


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    y can be anything and the inequality is still satisfied.
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