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Simpson's Rule

Simpson's Rule

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!

Simpson's Rule Definition and Formula for Area

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 Derivation

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

Simpson's Rule derivation using parabolic curve StudySmarterSimpson's Rule builds subintervals of quadratic curves between three points - StudySmarter Originals

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

Simpson's Rule derivation using parabolic curves StudySmarterSimpson's Rule builds a parabola from a group of three points and sums the area under each parabolic curve to approximate the total area under the curve - StudySmarter Originals

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.

Simpson's Rule Error

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.

Relative error

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.

Absolute error

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.

Error Bounds for Simpson's Rule

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.

Advantages and Limitations of Simpson's Rule

Advantages

  • 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!

Limitations

  • As three points are required to make a quadratic curve, Simpson's Rule requires an even number of subintervals n

  • Simpson's Rule performs with low accuracy for highly oscillating functions

Examples of Using Simpson's Rule to Estimate the Integral

Example 1

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.

Step 1: Find

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

Step 2: Plug in known values to Simpson's Rule

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.

Step 3: Consider the maximum error bound

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.

Simpson's Rule 4th derivative error bound calculation example StudySmarterThe 4th derivative of f(x) = 1/(1+x) attains a maximum at f(1) on the interval [1, 4] - StudySmarter Original

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.

Step 5: Find a minimum n such that the error is at most 0.001.

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.

Example 2

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

x-10123
f(x)108941

Step 1: Find

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

Step 2: Plug in known values to Simpson's Rule

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 - Key takeaways

  • 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 states

    where 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

Frequently Asked Questions about Simpson's 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 estimates the integral of f(x) on the interval [a, b] is (3((b - a)/n))(f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 2f(xn-2) + 4f(xn-1) + f(xn))

In Simpson's Rule, is an even number that represents the number of subintervals used to approximate the integral. If is not specified, use the error bound formula to find the minimum number of subintervals to ensure a low error.

Simpson's Rule is more accurate than the Trapezoidal Rule because it uses quadratic equations rather than linear equations to estimate the area of the subinterval.

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.

Final Simpson's Rule Quiz

Question

State Simpson's Rule in words.

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Answer

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 obtain an approximation for the total area under the curve

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Question

While the pattern of the Trapezoidal Rule coefficients is 1, 2, 2, ..., 2, 1, the pattern of Simpson's Rule coefficients is

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Answer

4, 2, 4, 2, ..., 4, 2

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Question

Simpson's Rule requires a(n)  ____ number of subregions

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Answer

even

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Question

Why is Simpson's Rule more accurate than the Trapezoidal Rule?

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Answer

Simpson's Rule divides the curve into quadratic curves, which are non-linear, while the Trapezoidal Rule divides the curve into trapezoids, which are linear

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Question

Simpson's Rule can find the exact answer to which kind of equations?

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Answer

Linear and quartic

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Question

What sort of functions does Simpson's Rule have a difficult time approximating?

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Answer

Highly oscillating functions

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