How Is the Error Term Derived for Simpson's Rule?

[makethings]

Hi can someone help me find a derivation of the error/remainder term for simpson's rule? None of the 3 math reference textbooks I have at home contain it.

 

[CRGreathouse]

http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html

 

[tacman]

Sure, I can help you with that. Simpson's rule is a numerical method used to approximate the value of a definite integral. It is based on dividing the interval of integration into smaller subintervals and using quadratic polynomials to approximate the function within each subinterval.

The error or remainder term for Simpson's rule can be derived using Taylor's theorem. This theorem states that any sufficiently smooth function can be approximated by a polynomial of degree n, where n is the number of terms in the polynomial.

In the case of Simpson's rule, the function is approximated by a quadratic polynomial within each subinterval. Let's say we have divided the interval of integration [a,b] into n subintervals, with each subinterval having a width of h = (b-a)/n. Then, the points where the function is evaluated are a, a+h, a+2h, …, b-h, b.

The quadratic polynomial that approximates the function within each subinterval is given by:

P(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2

Using Taylor's theorem, we can write the remainder term R(x) as:

R(x) = f(x) - P(x)

= f(x) - [f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2]

= f(x) - f(a) - f'(a)(x-a) - f''(a)(x-a)^2/2

Now, to find the error term for Simpson's rule, we need to integrate this remainder term over the entire interval [a,b]. This can be done by breaking up the integral into smaller subintervals and using the fact that the integral of a sum is equal to the sum of integrals.

∫a^b R(x) dx = ∫a^b [f(x) - f(a) - f'(a)(x-a) - f''(a)(x-a)^2/2] dx

= ∑(∫ai^ai+1 [f(x) - f(a) - f'(a)(x-a) - f''(a)(x-a)^2/2] dx)

= ∑(∫ai^ai+1 [f(x) - f(a) - f'(a)(x-a)] dx) - ∑(∫