# DISK_INTEGRALS Integrals Inside the Unit Disk in 2D

DISK_INTEGRALS is a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

The interior of the unit disk in 2D is defined by

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

The integrands are all of the form

```        f(x,y) = x^e1 * y^e2
```
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:

DISK_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 FORTRAN77 library which defines integrands for testing quadrature over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN77 library which defines integrands for testing quadrature over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2D.

DISK_RULE, a FORTRAN77 library which computes quadrature rules for the unit disk in 2D, that is, the interior of the circle of radius 1 and center (0,0).

HYPERBALL_INTEGRALS, a FORTRAN77 library which defines integrands for testing quadrature over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a FORTRAN77 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 FORTRAN77 library which defines integrands for testing quadrature over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a FORTRAN77 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 FORTRAN77 library which defines integrands for testing quadrature over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN77 library which defines integrands for testing quadrature over the interior of a triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN77 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:

• DISK01_AREA returns the area of the unit disk in 2D.
• DISK01_MONOMIAL_INTEGRAL returns monomial integrals in the unit disk in 2D.
• DISK01_SAMPLE uniformly samples points from the unit disk in 2D.
• I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
• MONOMIAL_VALUE evaluates a monomial.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.

Last revised on 08 January 2014.