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**.

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:

Where:

- a, b, = the endpoints of the closed interval [a, b].
- max|f′′(x)| = least upper bound of the second derivative.
- n = number of partitions (rectangles) used.

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

**Solution**:

Step 1: **Calculate the second derivative:** f′′ = 12x^{2}. 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′′ = 12x^{2}, 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:

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.

## 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 f^{4}: