# Extremly long Definite triple integral

1. May 9, 2013

### NamDogg

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. May 9, 2013

### Office_Shredder

Staff Emeritus
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?

3. May 9, 2013

### NamDogg

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)

4. May 9, 2013

### mathman

It depends very much on what the limits are. For example, you could change to spherical coordinates centered at (x',y',z').

5. May 9, 2013

### NamDogg

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