# Proving the Equality of Simpson's Rule using Trapezoid Rule Approximations

• KingKendrick
In summary, the conversation discusses the calculation of the Simpson's Rule approximation for a given integral. It is mentioned that there is a formula for Simpson's Rule, but there is a better way to calculate it using the Trapezoid Rule approximations. The formula for Simpson's Rule is given and it is suggested to verify its accuracy by substituting a random function. However, the professor clarifies that the solution should be done algebraically. The conversation ends with a request for the student to show their effort and calculations for S(16) and (4T(16)-T(8))/3.
KingKendrick

## Homework Statement

Integral of f(x)dx from a to b~ S(n) = f(x0) + 4(x1) +2(x2) + 4(x3) + ... 2(xn-2) + 4(xn-1) + f(xn))(Δx/3)

You can use the formula for Simpson's Rule given above, but here is a better way. If you already have the Trapezoid Rule approximations T(2n) and T(n), the next Simpson's rule approximation follows immediately with a simple calculation

S(2n) = (4T(2n) -T(n))/(3)

Verify that for n=8, the two forms of the Simpson's Rule are the same.

## The Attempt at a Solution

I tried to substitute a random function and attempt to go on from there but our professor said that we needed to solve this algebraically. Any help would be appreciated. Thanks!

You have to show effort. What did you get for S(16) and (4T(16)-T(8))/3?

Also you left out a bunch of f's in your formula for Simpson's rule.

## What is the Simpson Rule equation?

The Simpson Rule equation is a mathematical formula used to approximate the area under a curve by dividing it into smaller, simpler shapes and adding their areas together.

## Why is the Simpson Rule equation important?

The Simpson Rule equation is important because it provides a more accurate estimation of the area under a curve compared to other methods, such as the trapezoidal rule. It is also commonly used in scientific and engineering applications for numerical integration.

## How is the Simpson Rule equation derived?

The Simpson Rule equation is derived from Simpson's rule, which states that the area under the curve can be approximated by the average of the areas of the two trapezoids formed by dividing the curve into three equal parts. By using this rule iteratively, a more accurate estimation of the area can be obtained.

## What are the assumptions made in the Simpson Rule equation?

The Simpson Rule equation assumes that the curve is continuous, and that the interval between the points used for approximation is equally spaced. It also assumes that the curve is concave up or down, and that the function is integrable.

## What are the limitations of the Simpson Rule equation?

The Simpson Rule equation can only be used for functions that are integrable and have a known analytical solution. It also requires more computation compared to other methods, which can make it less efficient for large datasets.

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