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Area of a hyperbolic paraboloid contained within a cylinder

  1. Nov 2, 2009 #1
    I've posted on this before and have now realised I was doing it completely wrong before but's still bugging me. I have to find the area of the hyperbolic paraboloid z=xy contained within the cylinder x[tex]^{2}[/tex]+y[tex]^{2}[/tex]=1.

    I've parametrized x[tex]^{2}[/tex]+y[tex]^{2}[/tex]=1 into polar coordinates to give that dA=d[tex]\theta[/tex]dz

    I also know that 0[tex]\leq[/tex][tex]\theta[/tex][tex]\leq[/tex]2[tex]\pi[/tex]

    But I'm still having trouble finding the z limits for that part of the integration. Any help would be great. Thanks.
     
  2. jcsd
  3. Nov 2, 2009 #2

    LCKurtz

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    It isn't the cylinder you need to parameterize, it's the hyperbolic paraboloid. Incidentally, that part inside the cylinder looks just like a pringle. Try this parameterization:

    [tex]\vec{R}(r,\theta) = \langle r*cos(\theta), r*sin(\theta),r^2cos(\theta)sin(\theta)\rangle[/tex]

    with
    [tex]dS = |\vec{R}_r \times \vec{R}_\theta |dr d\theta[/tex]

    where [itex](r, \theta)[/itex] are the usual polar variables in the xy plane.
     
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