ball_integrals

ball_integrals, an Octave code 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

where the exponents are nonnegative integers. If any exponent is an odd integer, the integral will be zero. Thus, the "interesting" results occur when all exponents are even.

Licensing:

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

Languages:

ball_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

ball_integrals_test

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

Reference:

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.
monomial_value.m, evaluates a monomial.

Last revised on 11 October 2022.