# disk_integrals

disk_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of a disk of radius R centered at the origin in 2D.

The interior of the disk is defined by

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

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 MATLAB 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.

disk_monte_carlo, a MATLAB code which uses the Monte Carlo method to estimate the integral of a function over the interior of a disk of radius R centered at the origin.

disk_rule, a MATLAB code which computes quadrature rules for disk of radius R centered at the origin.

disk01_integrals, a MATLAB code which returns the exact value of the integral of a monomial over the interior of a unit disk centered at the origin 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_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.

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:

• disk_area.m, returns the area of a disk of radius R centered at the origin.
• disk_monomial_integral.m, returns the exact integral of a monomial over a disk of radius R centered at the origin.
• disk_sample.m, returns uniform sample points from disk of radius R centered at the origin.
• monomial_value.m, evaluates a monomial function.

Last revised on 04 November 2022.