Exact Simpson’s Rule

Prove that Simpson’s Rule is exact (no error) when approximating the definite integral of a linear function and a quadratic function.

Solution:-

Step1

Simpson’s rule

Three points integration rule derived using the method of undetermined coefficients.

Suppose that we add a quadrature point at the middle of the interval [a, b],

To avoid algebra,

substitute x = and define h =

so that the integral becomes

Since we have three unknowns,

we can make this formula exact for all quadratic functions;

so, let us use,

e.g., F ≡ 1, F ≡ u and F ≡ u 2 .

F ≡ 1 ⇒ = 2h = w1 + w2 + w3;

F ≡ u ⇒ = 0 = −hw1 + hw3 ⇒ w1 = w3;

F ≡ ⇒ = = w1 + w3 ⇒ w1 = w3 = ; w2 = 2h − w1 − w3 = 2h − h =

Hence we obtain the approximation

which...