BALL_INTEGRALS
Integrals Inside the Unit Ball in 3D

BALL_INTEGRALS is a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

The interior of the unit ball in 3D is defined by

        x^2 + y^2 + z^2 <= 1
      

The integrands are all of the form

        f(x,y,z) = x^e1 * y^e2 * z^e3
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

BALL_INTEGRALS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BALL_MONTE_CARLO, a MATLAB library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK01_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Reference:

1. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.

Source Code:

• ball01_monomial_integral.m, returns monomial integrals in the unit ball.
• ball01_sample.m, uniformly samples points from the unit ball.
• ball01_volume.m, returns the volume of the unit ball.
• i4vec_uniform_ab.m, returns a scaled pseudorandom I4VEC.
• monomial_value.m, evaluates a monomial.
• r8vec_uniform_01.m, returns a uniform pseudorandom R8VEC.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• ball_integrals_test.m, calls all the tests.
• ball_integrals_test01.m, tests the function on several monomials.
• ball_integrals_test.txt, the output file.

You can go up one level to the MATLAB source codes.