How to derive the quadrature weights for the trapezoid and the Simpsons rule?Trapeziod rule: ∫ a b f ( x ) ≈ b − a 2 [ f ( a ) + f ( b ) ]Simpson's rule: ∫ a b f ( x ) ≈ b − a 6 [ f ( a ) + 4 f ( ( a + b ) / 2 ) + f ( b ) ]

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How to derive the quadrature weights for the trapezoid and the Simpsons rule?Trapeziod rule:      ∫    a    b    f  (  x  )  ≈            b      −      a        2    [  f  (  a  )  +  f  (  b  )  ]Simpson&#039;s rule:      ∫    a    b    f  (  x  )  ≈            b      −      a        6    [  f  (  a  )  +  4  f  (  (  a  +  b  )      /    2  )  +  f  (  b  )  ]

Cortez Hughes · Accepted Answer

The Trapezoid and Simpson&#039;s rules are Newton-Cotes formulae. Derivation of the weights come from integrating Lagrange basis polynomials. Let   L  (  x  ) be a lagrange polynomial which interpolates   f  (  x  ) at   n  +  1 points,       x    0    ,      x    1    ,  .  .  .      x    n  . Then      ∫    a    b    f  (  x  )  d  x  ≈      ∫    a    b    L  (  x  )  d  x  =      ∫    a    b        ∑          i      =      0        n    f  (      x    i    )      l    i    (  x  )  d  x  =      ∑          i      =      0        n    f  (      x    i    )      ∫    a    b        l    i    (  x  )  d  x  ,and here       ∫          a        b        l    i    (  x  )  d  x is the weight for the   i-th interpolation point.

How to derive the quadrature weights for the trapezoid and the Simpsons rule? Trapeziod rule: <m

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