# hyperball_integrals

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

The interior of the unit hyperball in M dimensions is defined by

```        sum ( 1 <= i <= m ) x(i)^2 <= 1
```

The integrands are all of the form

```        f(x) = product ( 1 <= i <= m ) x(i)^e(i)
```
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.

### Languages:

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

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

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.

disk01_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

hexagon_integrals, a MATLAB code which returns the exact value of the integral of a monomial over the interior of a hexagon in 2D.

hyperball_monte_carlo, a MATLAB code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions.

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

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

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

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

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

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

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

wedge_integrals, a MATLAB code 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:

Last revised on 31 January 2019.