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Finding the Volume of a Region

  1. Dec 14, 2012 #1
    1. The problem statement, all variables and given/known data

    "Find the volume of the region bounded by (x2+y2+z2)2 = x"



    2. Relevant equations

    --> Need to get proper bounds on ρ, ∅, θ and evaluate triple integral.

    3. The attempt at a solution

    I have tried converting this into the spherical and cylindrical co-ordinates, but have yet to get any bounds at all...

    When using spherical co-ordinates, the equation turns into:

    ρ3 = sin∅cosθ

    which doesn't give bounds for anything..

    and similarly, if I use the cylindrical coordinates:

    (r2+z2)2 = rcosθ

    which doesn't give me bounds either..

    Am I missing something here?
     
    Last edited: Dec 14, 2012
  2. jcsd
  3. Dec 14, 2012 #2

    Mark44

    Staff: Mentor

    Sure it does. -1 ≤ sin(##\phi##) ≤ 1, and the same for cos(θ), so -1 ≤ ρ ≤ 1, although it might be that ρ ≥ 0 by convention.


     
  4. Dec 14, 2012 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If you actually try to do it using Mark44's comment in spherical coordinates you are going to get an awfully nasty integral. Use cylindrical. And here's a hint. Rotate the cylindrical coordinate system so the axis lies along the x-axis instead of the z-axis. Or you can think of this as rotating the volume. Now you want the volume of (r^2+z^2)^2=z. Makes life much easier.
     
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