Newton-Cotes formula using Hermite interpolation

[yon]

Homework Statement

Suppose we are given $$f_0, f_1, f'_0 f'_1$$ of an unknown function. We want to integrate the region between $$x_0 and x_1$$ using a Newton-Cotes formula.

Homework Equations

Wat is the 'standard' way to solve such a problem?

The Attempt at a Solution

I started with the standard Lagrange interpolation formula for 4 points x0, x1, x2 and x3, and I thought that if we let x2 -> x0 and x3 -> x1, then we remove the unknown x2 and x3, and we introduce the known f'_0 and f'_1. It would then be only a matter of integration to find the N.C. formula. However, I am not sure if this is the way to go to solve this problem?

 

[mighty2000]

Thank you for your question. The standard way to solve this problem is to use the Newton-Cotes formula for integration. This formula is based on the idea of approximating the function using a polynomial and then integrating the polynomial over the given interval. The formula involves evaluating the function at equally spaced points within the interval and then combining these values with a set of weights to approximate the integral.

In your case, you can use the given values of f_0, f_1, f'_0 and f'_1 to construct a polynomial of degree 3. Then, you can use the Newton-Cotes formula for integration to approximate the integral over the given interval. This will give you an approximate value for the integral.

Alternatively, you can also use the trapezoidal rule or Simpson's rule, which are specific cases of the Newton-Cotes formula, to approximate the integral. These methods may provide more accurate results depending on the nature of the function and the given data.

I hope this helps. Please let me know if you have any further questions.