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Simpson's rule

Simpson's rule is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Simpson's rule rather than just read about it. In short: In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads ∫ a b f ( x ) d x ≈ b − a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] . {\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].} In German a…

Simpson's rule — main illustration
Simpson's rule — illustration

Key takeaways

  • Simpson's rule belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Simpson's rule to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Simpson's rule from memory before moving on to harder problems.

Reference excerpt

In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads

∫ a b f ( x ) d x ≈ b − a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] . {\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].}

In German and some other languages, it is named after Johannes Kepler, who derived it in 1615 after seeing it used for wine barrels (barrel rule, Keplersche Fassregel). The approximate equality in the rule becomes exact if f is a polynomial up to and including 3rd degree. If the 1/3 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 1/3 rule. Points inside the integration range are given alternating weights 4/3 and 2/3. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve the order of the error. If the 3/8 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 3/8 rule. Simpson's 1/3 and 3/8 rules are two special cases of closed Newton–Cotes formulas. In naval architecture and ship stability estimation, there also exists Simpson's third rule, which has no special importance in general numerical analysis, see Simpson's rules (ship stability).

Simpson's 1/3 rule Simpson's 1/3 rule, also simply called Simpson's rule, is a method for numerical integration proposed by Thomas Simpson. It is based upon a quadratic interpolation and is the composite Simpson's 1/3 rule evaluated for n = 2 {\displaystyle n=2} . Simpson's 1/3 rule is as follows:

∫ a b f ( x ) d x ≈ b − a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] = 1 3 h [ f ( a ) + 4 f ( a + h ) + f ( b ) ] , {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right]\\&={\frac {1}{3}}h\left[f(a)+4f\left(a+h\right)+f(b)\right],\end{aligned}}}

where h = ( b − a ) / n {\displaystyle h=(b-a)/n} is the step size for n = 2 {\displaystyle n=2} . The error in approximating an integral by Simpson's rule for n = 2 {\displaystyle n=2} is

− 1 90 h 5 f ( 4 ) ( ξ ) = − ( b − a ) 5 2880 f ( 4 ) ( ξ ) , {\displaystyle -{\frac {1}{90}}h^{5}f^{(4)}(\xi )=-{\frac {(b-a)^{5}}{2880}}f^{(4)}(\xi ),}

… excerpt ends here. Continue reading the full article.

Illustrations

Simpson's rule: Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red).
Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red).
Simpson's rule: An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size
An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size
Simpson's rule: An animation showing how Simpson's rule approximation improves with more subdivisions.
An animation showing how Simpson's rule approximation improves with more subdivisions.
Simpson's rule: The standard normal distribution.
The standard normal distribution.

Worked examples

Example 1 — a first encounter with Simpson's rule

Start with the simplest possible case. Write down what Simpson's rule claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Simpson's rule before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Simpson's rule ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Simpson's rule

In research
Simpson's rule appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Simpson's rule in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Simpson's rule is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integral calculus, Numerical analysis, Numerical integration, so understanding it makes those chapters shorter.
In everyday life
Look for Simpson's rule outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Simpson's rule in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Simpson's rule means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Simpson's rule out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Simpson's rule in simple terms?

In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads ∫ a b f ( x ) d x ≈ b − a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] . {\displa…

Why does Simpson's rule matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Simpson's rule?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Simpson's rule.

Tags

  • Integral calculus
  • Numerical analysis
  • Numerical integration

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