You might like this one, I do not know if the entire link will fit, anyway

Journal of Approximation Theory Volume 104, Issue 1, Pages 1-182 (May 2000)

Uniqueness of the Gaussian Quadrature for a Ball Pages 21-44

B. Bojanova, 1 and G. Petrovab, 2

Department of Mathematics, University of Sofia, Boulevard James Boucher 5, 1164, Sofia, Bulgariaf1

Department of Mathematics, University of South Carolina, Columbia, South Carolina, 29208, U.S.A., f2 Received 8 June 1999; accepted 22 October 1999. ; Available online 26 March 2002.

Abstract

We construct a formula for numerical integration of functions over the unit ball in Image d that uses n Radon projections of these functions and is exact for all algebraic polynomials in Image d of degree 2n−1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadrature formula for a disk that is based on line integrals over n chords and integrates exactly all bivariate polynomials of degree 2n−1.

Author Keywords: Gauss quadrature formula; orthogonal polynomials; highest degree of precision

You might like this one, I do not know if the entire link will fit.

Wait, it is available for download from her website!

http://www.math.tamu.edu/~gpetrova/

Journal of Approximation Theory Volume 104, Issue 1,   (May 2000)

Uniqueness of the Gaussian Quadrature for a Ball Pages 21-44

Borislav Bojanov and Guergana Petrova

Department of Mathematics, University of Sofia, Boulevard James Boucher 5, 1164, Sofia, Bulgariaf1

Department of Mathematics, University of South Carolina, Columbia, South Carolina, 29208, U.S.A., f2 Received 8 June 1999; accepted 22 October 1999. ; Available online 26 March 2002.

Abstract

We construct a formula for numerical integration of functions over the unit ball in Image d that uses n Radon projections of these functions and is exact for all algebraic polynomials in Image d of degree 2n−1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadrature formula for a disk that is based on line integrals over n chords and integrates exactly all bivariate polynomials of degree 2n−1.

Author Keywords: Gauss quadrature formula; orthogonal polynomials; highest degree of precision

Different article by same people:

http://www.math.tamu.edu/~gpetrova/CAM7238.pdf