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Extremly long Definite triple integral

  1. May 9, 2013 #1
    Hi

    I am currently working on a project, and I need to calculate the definite triple integral of 1/|x+y+z|. i.e:

    int int int (1/sqrt((x-x')^2+(y-y')^2+(z-z')^2)) dx dy dz.
    I have solved the integral, and it has 48 terms if the limits are inserted (6 terms with 3 sets of upper and lower limits). The computing power to model with this huge expression is too much, so is there a trick or shortcut that outputs a simple answer, even if it is slightly approximated?

    Assistance would be greatly appreciated.
     
    Last edited: May 9, 2013
  2. jcsd
  3. May 9, 2013 #2

    Office_Shredder

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    I would argue that the function 1/|x+y+z| and the thing in your integral are different functions. I'm going to assume that your integral is correct

    What kind of region are you integrating over?
     
  4. May 9, 2013 #3
    Thank you for your reply.

    I am creating a geophysical inversion program, by which I treat one body as a set of vertical square columns. The only variables are then the density and depth of the individual columns. After defining the potential for one column, I can create an array of columns and find the forward model for the system. Hence my integral is:

    I= Gρ(∫∫∫1/|r-r0| dx dy dz)
    I= Gρ(∫∫∫1/sqrt((x-x')^2+(y-y')^2+(z-z')^2) dx dy dz))

    with respective x,y,z min and max constants as limits

    So I am integrating over real space.


    However, the integral has a huge amount of terms, so my modelling is quite slow, especially for systems with many vertical columns (large body)
     
  5. May 9, 2013 #4

    mathman

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    It depends very much on what the limits are. For example, you could change to spherical coordinates centered at (x',y',z').
     
  6. May 9, 2013 #5
    I don't see how spherical coordinates would be simpler for a rectangular prism array.
    My limits are:

    -t/2<x<t/2
    -t/2<y<t/2
    h<z<h+D

    t=thickness of the columns. Such that the area of the column is t^2
    h=measurement height above the z=0 plane. Such that g(0,0,0) is finite.
    D = depth of the column
     
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