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Order of integration.

  1. Dec 18, 2011 #1
    1. The problem statement, all variables and given/known data

    I have this question given:

    sdjm1.png


    2. Relevant equations



    3. The attempt at a solution

    So I used change of variables, fairly straightforward, I set

    a = x+y+z
    b = x+2y
    c = y - 3z

    computed the jacobian, and got the new ranges.

    Anyway, so the solution has the order of integration as
    int int int (sqrt a dc db da)

    why is it from dc db da? I was wrong at that part because i used da db dc, how do you find out what the order of integration should be?
     
  2. jcsd
  3. Dec 18, 2011 #2
    What you can do is to regard the boundary conditions as a system of linear equations. This will allow you to calculate values for x,y and z ( this is where the parallelpipes intersect ). Now stay in the same coordinate system and simply integrate over dxdydz using the values obtained earlier - make sure they are oriented correctly, i.e. the signs are correct. The actual integration should then be straightforward.
     
  4. Dec 18, 2011 #3
    I get what you are saying, but its volume, so why does it matter if you integrate with respect to z, y and x rather than in the other order? (assuming the restrictions are correct for each order).

    Change of variable has to be used for this question because it makes it a lot easier. What I want to know though is the order of integration when you perform the change of variable.
     
  5. Dec 18, 2011 #4
    Well, the end result, if done correctly, will be the same for each method.
    I personally think that this particular integral is actually much easier to calculate if you stay in the [x,y,z] space; the order of variables doesn't need to change either, in fact the order doesn't even matter so long as you have the integration limits right.
     
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