numerical integration - Unexpected result of summation

I wrote a small module that gives me an incorrect output-set. It should be a single number!

I don't understand what went wrong.

This is the form of summation used: $$\frac{1}{2} (b-a) \sum_{i=1}^n w_i\,f\left(\frac{b-a}{2}x_i+\frac{a+b}{2}\right)$$ where ${x,w}$ are generated by GaussianQuadratureWeights command:

  Needs["NumericalDifferentialEquationAnalysis`"]
    GaussLegendreQuadrature[a_, b_, n_] := Module[{weights, i},

       (* GaussianQuadratureWeights[n, a, b] gives a list of the
          n pairs {Subscript[x, i], Subscript[w, i]} of the elementary
          n-point Gaussian formula for quadrature on the interval a to b,
          where Subscript[w, i] is the weight of the abscissa
          Subscript[x, i].*)

       weights = GaussianQuadratureWeights[n, -1, 1];
       Return[(b - a)/2 \!\(
    \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(
    \*SubscriptBox[\(weights\), \(i, 2\)]\ f[

    \*FractionBox[\(a + b\), \(2\)] + 
    \*FractionBox[\(b - a\), \(2\)] 
    \*SubscriptBox[\(weights\), \(i, 1\)]]\)\)]];

    f[x_] := Sin[x];
    GaussLegendre[0, 1, 8]

Answer

I can see two problems : you need to include the function as an argument to GaussLegendreQuadrature and you need to call it with the correct name.

GaussLegendreQuadrature[a_, b_, n_, f_] := Module[{weights, i},
  weights = GaussianQuadratureWeights[n, -1, 1];

  (b-a)/2 * Sum[weights[[i, 2]] f[(a + b)/2 + (b - a)/2 weights[[i, 1]]], {i, 1,n}]]    

GaussLegendreQuadrature[0, 1, 8, Sin]

(* 0.459698 *)