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Errors in the Trapezoidal Rule and Simpson’s Rule

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The trapezoidal rule and Simpson’s rule are an approximate way to calculate the area under a curve (i.e. a definite integral). It’s possible to calculate how well these rules approximate the area with the Error Bounds formula.

numerical quadrature

The trapezoid rule with n = 6 partitions.



The “error” is the difference between the actual “true” value and the approximation. Errors in the trapezoidal rule and Simpson’s rule can be calculated with a couple of straightforward formulas; These are useful when we want to increase the accuracy of an approximation. Increasing the number of partitions leads to better and better approximations: the following formulas give you a way to quantify those errors.

Errors in the Trapezoidal Rule and Simpson’s Rule: Formula

1. Error Bounds Formula for Trapezoidal Rule

The error formula for the trapezoidal rule is:
error for trapezoidal rule 2

Where:

Example Question: What is the error using the trapezoidal rule for the function f(x) = x4 with 4 intervals on [0, 4]?

Solution:


Step 1: Calculate the second derivative: f′′ = 12x2. If the second derivative is not a continuous function, you cannot use the formula.

Step 2: Find the least upper bound (the “max”) of the second derivative on the interval (for this example, the interval is [0, 4]. You can do this in two ways:


  • Look at a graph and locate the max on the interval, or
  • Find the critical numbers and evaluate the function for those numbers (including at the endpoints).

Looking at a graph of f′′ = 12x2, we can see that the max value is f(x) = 192.

Step 3: Set up the formula and solve: Plugging in our numbers, we get:
error for trapezoidal rule solution

Where:

  • a, b are given in the question as 0, 4,
  • n = 4 (from the question)
  • max|f′′(x)| = LUB from Step 3.

The error between the Trapezoid rule and definite integral is 64. Increasing the number of partitions “n” will result in better approximations.

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2. Error Bounds Formula for Simpson’s Rule

The error formula for Simpson’s rule is similar. The main difference is that it uses the max of the fourth derivative f4:
error for simpsons rule 2

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CITE THIS AS:
Stephanie Glen. "Errors in the Trapezoidal Rule and Simpson’s Rule" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/errors-in-the-trapezoidal-rule-and-simpsons-rule/
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