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Simpson's Rule

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In The Trapezoidal Rule article, we discussed the method of adding up areas of trapezoids to approximate areas under a function. In most cases, using trapezoids results in less error than using rectangles. But, could there be an even more accurate method than the trapezoidal rule? The answer is yes! Like the Trapezoidal Rule, Simpson's Rule is another numerical integration technique used to approximate an integral that may be too difficult to compute directly. Unlike the Trapezoidal Rule, Simpson's Rule uses quadratic polynomial approximation, making it a more accurate integral estimation technique. Let's take a deeper look as to what we mean!

Before we get into how this technique is used in practice, let's define this rule!

**Simpson's Rule **is an integral approximation technique that divides the area under the curve into smaller curves. The area under each smaller curve is added together to obtain an approximation for the total area under the curve.

Simpson's Rule uses the simple fact that we can build a quadratic equation from any three points. Like the Trapezoidal Rule, Simpson's Rule creates *n *subintervals. For each pair of consecutive subintervals and , Simpson's Rule builds a quadratic equation of the form through the three points .

Using the equation for a quadratic curve, we can find the area under the curve that passes through the points . Letting and and integrating over the interval we have

Since the points and are all on the parabola, we can say

Note that

So, we can say that the area under the parabola is

However, when applying Simpson's Rule, we usually use more than just one parabolic curve. Essentially, we end up "integrating" a piecewise quadratic function. So, our area equation becomes

where is the distance between each

Simplifying this equation, we get an approximation for the definite integral of a function *f(x) *called Simpson's Rule, which states

where *n *is the number of subintervals, , and.

Just like in the Trapezoidal Rule, increasing *n *will also increase the accuracy of the integral approximation.

Unlike the Trapezoidal Rule, where we can determine whether our approximation is an over or underestimate based on the curve's concavity, there are no clear indicators for over or underestimation using Simpson's Rule. However, we can use the relative and absolute errors to find out more about how our estimation compares to the actual value.

We compute the error of a Simpson's Rule computation by using the relative error formula:

where is the Simpson's Rule approximation of the integral.

In addition to relative error, the absolute error of our approximation using Simpson's rule can be calculated using the formula for absolute error:

However, as mentioned in The Trapezoidal Rule article, the integral cannot always be computed exactly.

Like the Trapezoidal Rule, Simpson's Rule has an error-bound formula, which describes the maximum possible error of our approximation. For Simpson's Rule, the error-bound formula is

for

where is the exact error for Simpson's Rule and is the fourth derivative of *f(x)*. K is the fourth derivative's maximum value on the interval .

The use of the error bound will make more sense once we work through some examples.

Simpson's Rule is more accurate than the Trapezoidal Rule

Simpson's Rule is exact for cubic functions (of the form ), quadratic functions, and linear functions

Why is the Simpson's Rule exact for functions of order 3 and less? The fourth derivative of a function of order 3 and less is 0!

As three points are required to make a quadratic curve, Simpson's Rule requires an

**even***n*Simpson's Rule performs with low accuracy for highly oscillating functions

**Use Simpson's Rule to estimate the integral with n = 6. Then, find the minimum number of subintervals n to ensure a maximum error of 0.001.**

Luckily, the process of Simpson's Rule is very similar to the Trapezoidal Rule.

Plugging in our given interval and **even **number of *n *subregions:

All we need to do from here is plug our known values into the Simpson's Rule formula. Since our interval is [1, 4] and the problem asks us to use n = 6 subregions, meaning each subregion has a width of units.

Notice how the pattern of the coefficients is 1, 4, 2, 4, 2, ..., 2, 4, 1.

Let's use our error bound formula to see exactly how much of an overestimate our approximation is.

In the error bound formula , our only unknown value is *K*. However, we can use the fourth derivative of *f(x) *to find *K:*

To find *K*, we have to consider where * *will reach its maximum value on the interval [1, 4]. We can graph to find the maximum value on the interval.

We can see that the fourth derivative reaches its largest value at . Now that all the values ofare known, we can plug in to find our bound.

At most, the error of our estimation is 0.00078.

Clearly our error for *n *= 6 is less than 0.001. However, let's find the *minimum* *n* necessary to achieve an error of at most 0.001.

We let *n *be out unknown in our error bound.

We can disregard the second solution, , in this situation because we cannot have a negative amount of subregions. So, to ensure that our error is at most 0.001, we must use at least 6 subregions.

If you end up with an odd number of subregions, you must round up to an even number, as required for the Simpson's rule.

Use Simpson's Rule to approximate the area under the curve of *f(x) *given in the table below with n = 4.

x | -1 | 0 | 1 | 2 | 3 |

f(x) | 10 | 8 | 9 | 4 | 1 |

Plugging in our given interval and **even** number of *n *subregions:

From here, all we need to do is plug our known values into the Simpson's Rule formula. Since our interval is [-1, 3] and the problem asks us to use n = 4 subregions, meaning each subregion has a width of 1 unit.

**Simpson's Rule**is an integral approximation technique that divides the area under the curve into smaller curves and sums the area under each smaller curve to approximate the total area under the curve- For approximating the definite integral of a function
*f(x)*, Simpson's Rule stateswhere

*n*is the number of subintervals, , and Simpson's Rule builds a quadratic curve out of each sequential subinterval

- We can use an error-bound formula to tell us the maximum possible error of our approximation
For Simpson's Rule, the error-bound formula is

for

where is the exact error of Simpson's Rule and is the fourth derivative of

*f(x)*

- While Simpson's Rule is more accurate than the Trapezoidal Rule, Simpson's Rule requires an even number of
*n*subregions

*f(x) *on the interval [a, b] is (3((b - a)/n))(f(x_{0}) + 4f(x_{1}) + 2f(x_{2}) + 4f(x_{3}) + 2f(x_{4}) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_{n}))

*n *is an even number that represents the number of subintervals used to approximate the integral. If *n *is not specified, use the error bound formula to find the minimum number of subintervals *n *to ensure a low error.

More about Simpson's Rule

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