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Trapezoidal rule

Trapezoidal rule is a science 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 Trapezoidal rule rather than just read about it. In short: In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule) is a technique for numerical integration, i.e. approximating the definite integral: ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area. This is easily calc…

Trapezoidal rule — main illustration
Trapezoidal rule — illustration

Key takeaways

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

Reference excerpt

In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule) is a technique for numerical integration, i.e. approximating the definite integral:

∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.}

The trapezoidal rule works by approximating the region under the graph of the function

f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area. This is easily calculated by noting that the area of the region is made up of a rectangle with width ( b − a ) {\displaystyle (b-a)} and height f ( a ) {\displaystyle f(a)} , and a triangle of width ( b − a ) {\displaystyle (b-a)} and height f ( b ) − f ( a ) {\displaystyle f(b)-f(a)} . Therefore,

∫ a b f ( x ) d x ≈ ( b − a ) ⋅ f ( a ) ⏟ Area of rectangle + 1 2 ( b − a ) ⋅ [ f ( b ) − f ( a ) ] ⏟ Area of triangle = ( b − a ) ⋅ ( f ( a ) + 1 2 f ( b ) − 1 2 f ( a ) ) = ( b − a ) ⋅ ( 1 2 f ( a ) + 1 2 f ( b ) ) = 1 2 ( b − a ) [ f ( a ) + f ( b ) ] . {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx \underbrace {(b-a)\cdot f(a)} _{\text{Area of rectangle}}+\underbrace {{\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)]} _{\text{Area of triangle}}\\&=(b-a)\cdot \left(f(a)+{\tfrac {1}{2}}f(b)-{\tfrac {1}{2}}f(a)\right)\\&=(b-a)\cdot \left({\tfrac {1}{2}}f(a)+{\tfrac {1}{2}}f(b)\right)\\&={\frac {1}{2}}(b-a)[f(a)+f(b)].\end{aligned}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Trapezoidal rule: The function f(x) (in blue) is approximated by a linear function (in red).
The function f(x) (in blue) is approximated by a linear function (in red).
Trapezoidal rule: An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases
An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases
Trapezoidal rule: Illustration of "chained trapezoidal rule" used on an irregularly spaced partition of [ a , b ] {\displaystyle [a,b]}
Illustration of "chained trapezoidal rule" used on an irregularly spaced partition of [ a , b ] {\displaystyle [a,b]}
Trapezoidal rule: An animation showing how the trapezoidal rule approximation improves with more strips for an interval with a = 2 {\displaystyle a=2} and b = 8 {\displaystyle b=8} . As the number of intervals N {\displaystyle N} increases, so too does the accuracy of the result.
An animation showing how the trapezoidal rule approximation improves with more strips for an interval with a = 2 {\displaystyle a=2} and b = 8 {\displaystyle b=8} . As the number of intervals N {\displaystyle N} increases, so too does the accuracy of the result.

Worked examples

Example 1 — a first encounter with Trapezoidal rule

Start with the simplest possible case. Write down what Trapezoidal rule claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Trapezoidal 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 Trapezoidal 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 Trapezoidal rule

In research
Trapezoidal rule appears in science 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 Trapezoidal 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
Trapezoidal rule is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical integration, so understanding it makes those chapters shorter.
In everyday life
Look for Trapezoidal 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 Trapezoidal rule in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Trapezoidal 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 Trapezoidal rule out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Trapezoidal rule in simple terms?

In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule) is a technique for numerical integration, i.e. approximating the definite integral: ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} The trapezoidal rule works by approximating the region u…

Why does Trapezoidal rule matter?

Because it connects several science 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 Trapezoidal 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 Trapezoidal rule.

Tags

  • Numerical integration

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