Opuscula Math. 28, no. 2 (2008), 109-121

Opuscula Mathematica

# A note on a family of quadrature formulas and some applications

Bogusław Bożek
Wiesław Solak
Zbigniew Szydełko

Abstract. In this paper a construction of a one-parameter family of quadrature formulas is presented. This family contains the classical quadrature formulas: trapezoidal rule, midpoint rule and two-point Gauss rule. One can prove that for any continuous function there exists a parameter for which the value of quadrature formula is equal to the integral. Some applications of this family to the construction of cubature formulas, numerical solution of ordinary differential equations and integral equations are presented.

Keywords: quadrature and cubature formulas, numerical integration.

Mathematics Subject Classification: 65D30, 65D32, 65L05, 65L06, 65R20.

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