proof of Simpson’s rule

We want to derive Simpson’s rule for


We will use Newton and Cotes formulas for n=2. In this case, x0=a, x2=b and x1=(a+b)/2. We use Lagrange’s interpolation formula to get a polynomialPlanetmathPlanetmath p(x) such that p(xj)=f(xj) for j=0,1,2.

The corresponding interpolating polynomial is


and thus


Since integration is linear, we are concerned only with integrating each term in the sum. Now, taking xj=a+hj where j=0,1,2 and h=|b-a|/2, we can rewrite the quotients on the last integral as


and if we calculate the integrals on the last expression we get


which is Simpson’s rule:

Title proof of Simpson’s rule
Canonical name ProofOfSimpsonsRule
Date of creation 2013-03-22 14:50:25
Last modified on 2013-03-22 14:50:25
Owner drini (3)
Last modified by drini (3)
Numerical id 4
Author drini (3)
Entry type Proof
Classification msc 65D32
Classification msc 41A55
Classification msc 26A06
Classification msc 28-00