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Forced to use symmetry to solve this double integral?

  1. Dec 4, 2013 #1
    1. The problem statement, all variables and given/known data

    http://i.imgur.com/d4ViHux.png

    2. Relevant equations



    3. The attempt at a solution

    The author writes: "Now, using symmetry, we have..."

    But what symmetry does the author use? Also, I got the integral as shown in the remark but why is it wrong?
     
  2. jcsd
  3. Dec 4, 2013 #2

    cepheid

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    The symmetry that's used is only to integrate over only one semi-circle in the xy-plane, and not the other. That's why θ only ranges from 0 to π/2, not all the way to π. That's also why there is a factor of 2 in front of the integral: because the integral over one semi-circle should be exactly equal to the integral over the other one. This is because the portion of the hemisphere is that is above one semi-circle is equal in volume to the portion that is above the other: one portion is just the reflection of the other one across the y-axis. That is the symmetry.
     
  4. Dec 4, 2013 #3
    Alright, but why is the integral shown in the remark incorrect?
     
  5. Dec 4, 2013 #4

    vela

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    What did you get when you evaluated it? The integral isn't incorrect, but you have to be careful when evaluating it to get the correct result.
     
  6. Dec 4, 2013 #5

    haruspex

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    You have to be careful whenever you 'execute' a square root. √(x2) is not the same as x.
     
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