# Extremly long Definite triple integral

• NamDogg
In summary: I don't see how spherical coordinates would be simpler for a rectangular prism array. Assistance would be greatly appreciated.
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:
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?

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)

NamDogg said:
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').

mathman said:
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

## What is an extremely long definite triple integral?

An extremely long definite triple integral is a type of mathematical calculation used in multivariable calculus. It involves integrating a function over a three-dimensional region, with each variable having a specific range of values.

## Why is it called a "definite" triple integral?

The term "definite" refers to the fact that the integral has specific limits of integration for each variable, rather than indefinite limits as in an indefinite integral. This allows for a precise calculation of the integral's value.

## What are the applications of extremely long definite triple integrals?

Extremely long definite triple integrals are commonly used in physics, engineering, and other branches of science to solve problems involving three-dimensional objects or systems. They are also used in economics and statistics to model and analyze complex systems.

## How do you solve an extremely long definite triple integral?

Solving an extremely long definite triple integral involves breaking it down into smaller, more manageable integrals using the rules of integration. This can be done by integrating one variable at a time, or by using advanced techniques such as change of variables.

## What are some common challenges when working with extremely long definite triple integrals?

Some common challenges include determining the appropriate limits of integration, choosing the most efficient method of integration, and dealing with complex functions or regions of integration. It is also important to check for symmetry and ensure proper notation is used throughout the calculation.

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