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Homework Help: Volume Integral

  1. Apr 15, 2005 #1


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    "Find the volume of the region enclosed between the survaces [tex]z=x^2 + y^2 [/tex] and [tex]z=2x[/tex]"

    I figured that the simplest way of doing this was to switch to a cylindrical co-ordinate system. Can someone check that the limits of integration are then
    [tex]-\frac{\pi}{2}\leq \theta \leq\frac{\pi}{2}[/tex]
    [tex]0\leq\ r \leq 2\cos(\theta)[/tex]
    [tex]r^2\leq\ z \leq 2 r \cos(\theta) [/tex]
    (and the jacobian being r)

    Thanks greatfully
  2. jcsd
  3. Apr 15, 2005 #2


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    Science Advisor

    z= 2x is a plane and forms the top of the figure. You are correct that you should use cylindrical coordinates. But z= x2+ y2 is a paraboloid. it's interesection with z= 2x is z= 2x= x2[/wsup]+ y2 or x2- 2x+ 1 + y2= (x-1)2+ y2 = 1 which, projected down in to the xy-plane is the circle with center (1,0) and radius 1. In cylindrical coordinates, x2+ y2= 2x is r2= 2rcos θ or
    r= 2 cos &theta. THAT is the fomula you need.
  4. Apr 15, 2005 #3


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    Unless I'm missing something (which is entirely possible), is that not what I have?
  5. Apr 15, 2005 #4


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    Homework Helper

    Yes, I believe your limits of integration are correct.
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