# SPHERE_INTEGRALS Integrals on the Surface of the Unit Sphere in 3D

SPHERE_INTEGRALS is a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

The surface of the unit sphere in 3D is defined by

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

The integrands are all of the form

```        f(x,y,z) = x^a y^b z^c
```
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:

SPHERE_INTEGRALS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

BALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate the integral of a function over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN90 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.

### List of Routines:

• I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
• MONOMIAL_VALUE evaluates a monomial.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• SPHERE01_AREA returns the area of the surface of the unit sphere in 3D.
• SPHERE01_MONOMIAL_INTEGRAL: integrals on surface of the unit sphere in 3D.
• SPHERE01_SAMPLE samples points from the surface of the unit sphere in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 06 January 2014.