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Multivariable Calculus 3D co-ord. system help

  1. Sep 12, 2009 #1
    1. Write inequalities to describe the region: The solid cylinder that lies on or below the plane z=8 and on or above the disk in the xy-plane with a center the origin and radius 2.


    I don't understand because i'm using stewart's calculus i have no idea what the equation for a cylinder is? they only show you the eqn of spheres (x-h)^2+(y-k)^2+(z-l)^2 = r^2
     
  2. jcsd
  3. Sep 12, 2009 #2
    In three dimensions, the general equation for a cylinder with cross section ellipse is x2/a2 + y2/b2 = 1.
    If the cross sections are circles then you can say x2 + y2 = r2
     
  4. Sep 12, 2009 #3

    I understand that, however i don't understand how i can have the radius of 2 in this inequality if the cylinder fills the z axis from z=0 to z=8 how do i show this? is it 0<x^2+y^2<8?
    where do i show the radius is 2 in this inequality? because the height of the cylinder is 8. the radius is 2. i cant show both? please help it would be greatly appreciated as there is a test on this material soon
     
  5. Sep 12, 2009 #4

    gabbagabbahey

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    The cylinders axis is coincident with the z-axis, so the fact that it lies between z=0 and z=8 tell you its height is 8 units.

    You need two inequalities to describe the region enclosed by this cylinder

    Start with a simpler problem...what inequality would represent the area enclosed by the circle [itex]x^2+y^2=4[/itex]?
     
  6. Sep 13, 2009 #5

    -2 < x^2+y^2 < 2 on the x-axis but i don't understand the z-axis can you just tell me how to incorperate the z=0 to z=8 in this? it doesnt fit in.
     
  7. Sep 13, 2009 #6

    gabbagabbahey

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    No, the smallest possible value of [itex]x^2+y^2[/itex] is zero (remember, [itex]x^2+y^2[/itex] represents the square of distance of the point(x,y) from the z-axis)...the furthest a point inside the circle [itex]x^2+y^2=4[/itex] can be from the z-axis is if it lies on the perimeter of the circle (2 units away from the z-axis), and the closest it can be to the z-axis is if it actually lies on the z-axis....so [itex]0\leq x^2+y^2\leq 4[/itex]...make sense?

    Points inside the cylinder will satisfy the same inequailty right?

    They will also be somewhere between z=0 and z=8 right?

    So...[itex]0\leq x^2+y^2\leq 4[/itex] and [itex]0\leq z\leq 8[/itex] describes the cylinder....make sense?
     
  8. Sep 13, 2009 #7
    yes, that makes sense because 0<x^2+y^2<4 describes the circle portion of the cylinder, but what is the point of saying it's greater than 0 if 0 is the lowest and would just confuse more could you also say x^2+y^2=4, 0<z<8?. because if it was 1<x^y+y^2<4 it would be a cylinder with a hole kind of like a toilet paper roll? but for cylinders do you always have to state the inequality for z independently because otherwise you would simply get a circular plane in R^3? thank you so much for this i really do appreciate it i was so confused but you cleared it up i think!
     
    Last edited: Sep 13, 2009
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