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.
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