Triple integrals in a specific region of space

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Integrate a polynomial over a polyhedron

Overview

The question is about integrating a polynomial over a region defined by linear inequalities.

Such a region is called a polytope or a polyhedron.

In Sage one can construct such regions using the Polyhedron class.

This class provides a method integrate which allows, when the latte_int optional package is installed, to integrate a polynomial function over a polyhedron.

In a way, the trick is: rather than start from the function integrate and somehow specify a region, we fist define a polyhedron and then use the method integrate to compute the integral of a polynomial over that polyhedron.

Steps

The region is defined by five inequalities:

x + 2 y + 3 z < 2
-1 < x
x < y
y < z
z < 1

and the function to integrate is

x^2 + 2 y z

The five inequalities can be rewritten as

2 + (-1) * x + (-2) * y + (-3) * z > 0
1 + (+1) * x                       > 0
0 + (-1) * x + (+1) * y            > 0
0            + (-1) * y + (+1) * z > 0
1                       + (-1) * z > 0

Encode these inequalities in Sage as

sage: ieqs = [[2, -1, -2, -3],
....:         [1,  1,  0,  0],
....:         [0, -1,  1,  0],
....:         [0,  0, -1,  1],
....:         [1,  0,  0, -1], ]

Create a polyhedron from those inequalities:

sage: P = Polyhedron(ieqs=ieqs)

Define polynomial variables and a polynomial:

sage: x, y, z = polygens(QQ, names='x, y, z')
sage: f = x^2 + 2*y*z

Integrate that polynomial over the polyhedron we defined:

sage: P.integrate(f)
53833/151875

This last command requires the latte_int optional package for Sage to be installed.

The latte_int optional package

To install that package, if you installed Sage from source or from binaries downloaded from the Sage website, run the following command in a terminal:

$ sage -i latte_int

Documentation

• SageMath documentation: integrate method for the Polyhedron class

Related

• Sage Trac ticket 22824: Add "see also" section to "integrate over a polytope"