# Extremly long Definite triple integral

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:

Office_Shredder
Staff Emeritus
Gold Member
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?

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)

mathman
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.

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

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

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