# Trapezoidal, simpsons rule, and higher order approximations

• okkvlt
In summary, the speaker was able to prove the trapezoidal rule and Simpson's rule by using matrices to determine coefficients for the approximating polynomial. They were amazed to see how the terms cancelled out and simplified the formulas. However, they wondered if approximating each step with a cubic equation would simplify the process. They plan to research Newton-Cotes formulae and Gaussian quadrature to see if these numerical techniques could simplify their method.
okkvlt
hi. i was able to prove the trapezoidal rule and simpsons rule. (basically i used matrices to determine the coefficients m and b for mx+b when proving the trapezoidal rule and a,b,c for ax^2+bx+c such that the points coincide, then i integrated the approximating polynomial) the amount of number-crunching and expanding products of linear terms was probably the most work I've ever done, but i was amazed to see terms cancel out to yeild vastly simplified formulas. but i was wondering, suppose i approximated each step with suppose, a cubic equation. i know how to do this, but after setting up the 4x4 matrix i quickly realized that this would be extremely mundane due to all the algebraic manipulations. so before i try deriving a numerical method that uses cubics to approximate the curve pieces, i want to know whether it will simplify. will it simplify?

## 1. What is the trapezoidal rule?

The trapezoidal rule is a method for approximating the definite integral of a function by dividing the area under the curve into trapezoids and summing their areas. It is based on the idea that the area under the curve can be approximated by the sum of the areas of these trapezoids.

## 2. How does the trapezoidal rule work?

The trapezoidal rule works by first dividing the interval of integration into smaller subintervals. Then, the area of each trapezoid is calculated using the height of the function at the endpoints of the subinterval. Finally, these areas are summed to approximate the area under the curve.

## 3. What is Simpsons rule?

Simpsons rule is a method for approximating the definite integral of a function by using quadratic polynomials to connect three points on the curve. It is more accurate than the trapezoidal rule and can be thought of as a combination of the trapezoidal rule and the midpoint rule.

## 4. How does Simpsons rule differ from the trapezoidal rule?

Simpsons rule differs from the trapezoidal rule in that it uses quadratic polynomials instead of straight lines to approximate the curve. This results in a more accurate approximation of the integral. Additionally, Simpsons rule uses more data points and therefore requires more calculations compared to the trapezoidal rule.

## 5. What are higher order approximations?

Higher order approximations refer to methods that use more data points and more complex mathematical techniques to approximate integrals. These methods, such as Simpson's 3/8 rule and Gaussian quadrature, are more accurate than the trapezoidal and Simpson's rule, but also require more computational effort.

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