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Math Help - Cylindrical Coordinates

  1. #1
    Member Em Yeu Anh's Avatar
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    Question Cylindrical Coordinates

    Describe the solid by inequalities in cylindrical coordinates which is enclosed by z=0, z=x+y+1 and the two cylinders x^2+y^2=4, x^2+y^2=9

    Just need someone to quickly verify this,
    I have
    2 \leq r \leq 3 ,
    0 \leq \theta \leq 2\pi and
    0 \leq z \leq x+y+1 (not sure how else to state that one)
    is this correct?
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  2. #2
    Super Member
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    I would draw it first and then work backwards. Now cylindrical coordinates are {r,t,z} so you have r right but from the plot below, z will depend on x=r\cos(t) and y=r\sin(t) and note as I increase z, the angle that t goes through changes as a function of z. But we can find that angle from solving the simultaneous equation:

    x^2+y^2=4

    x+y+1=z

    for constant z then obtain the two angles as (t_1(z),t_2(z)) then I think it would go something like:

    2\leq r \leq 3

    t_1(z)\leq t \leq t_2(z)

    0\leq z\leq r\cos(t)+r\sin(t)+1
    Attached Thumbnails Attached Thumbnails Cylindrical Coordinates-cylinder-space.jpg  
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  3. #3
    Member Em Yeu Anh's Avatar
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    Thank you very much for providing a picture, I made a careless mistake of assuming that the plane did not cut the object through the xy-plane. However, it isn't clear to me that theta is changing with respect to z from examining the plot at different angles.

    I'm still a little bit confused on how to go about solving that. Would you mind explaining to me the choice of equating x+y+1=z with x^2+y^2=4 rather than x^2+y^2=9?
    Last edited by Em Yeu Anh; March 22nd 2010 at 06:27 PM.
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