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Find The Volume of A Solid

  1. Dec 1, 2011 #1
    1. The problem statement, all variables and given/known data

    find the volume of the solid bounded by the graphs of the given equations:

    (the first octant)

    2. Relevant equations

    V=[itex]\int[/itex][itex]\int[/itex] rdrd∅

    3. The attempt at a solution

    So, I've been having trouble deciding what to integrate from and to. I converted the z equations into polar coordinates, so, z=rsin and z=0

    [itex]\int[/itex][itex]^{2\pi}_{0}[/itex][itex]\int[/itex][itex]^{1+cos∅}_{0}[/itex] (rsin)rdrd∅

    Mod note: revised LaTeX
    [tex]\int_0^{2\pi} \int_0^{1 + cos(\theta)} rsin(\theta) r~dr~d\theta[/tex]

    I found, from the first integration, [2∏]\int[/0][itex] r^(3)sin∅/3 d∅ and I need to plug in (1+cos∅) and 0 for r....but, this seems WAY too complicated having a trig function to the 3rd power.

    I'm not even sure if I have the beginning correct. Can anyone help me out?
    Last edited by a moderator: Dec 1, 2011
  2. jcsd
  3. Dec 1, 2011 #2


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    Staff Emeritus
    Science Advisor

    In three dimensions, the first quadrant has x ≥ 0, y ≥ 0, z ≥ 0, and think about the limits of ∅ for x ≥ 0, y ≥ 0.

    Think about this - http://www.wolframalpha.com/input/?i=polar+plot+r=1+cos+theta (don't worry about the angle being theta)

    And do you want to use cylindrical or spherical coordinates?
    Last edited: Dec 1, 2011
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