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11.9PureCore

The Trapezium Rule

When an integral cannot be found exactly, the trapezium rule estimates the area by splitting it into strips and approximating each by a trapezium. More strips give a better estimate.

24 min Integration
Edexcel A level Maths: 11.9 The Trapezium RuleCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • State the trapezium rule
  • Find the strip width h
  • Apply the rule to estimate an integral
  • Judge whether it over- or under-estimates
1

The rule

With nn strips of width h=b−anh = \dfrac{b - a}{n} and ordinates y0,y1,…,yny_0, y_1, \dots, y_n: ∫aby dx≈h2[y0+yn+2(y1+⋯+yn−1)]\displaystyle\int_a^b y\,dx \approx \tfrac{h}{2}\big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\big].

∫aby dx≈h2[y0+yn+2(y1+⋯+yn−1)]\int_a^b y\,dx \approx \tfrac{h}{2}\left[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\right]
Trapezium rule.
Example 1How many ordinates for 4 strips?
1Number of ordinates = strips + 1.
24+1=54 + 1 = 5 ordinates.
Answer
5 ordinates (y₀ to y₄).

Tip — The first and last ordinates have weight 1; all the middle ones have weight 2.

2

Over- or under-estimate

For a curve that is convex\textbf{convex} (bends upward) the trapezia lie above the curve, so the rule over\textbf{over}-estimates; for a concave curve it under-estimates.

Formula recap

h=b−anh = \frac{b - a}{n}
Strip width.
h2[y0+yn+2(… )]\tfrac{h}{2}[y_0 + y_n + 2(\dots)]
Trapezium rule.

Common mistakes to avoid

Using n ordinates for n strips.
There are n + 1 ordinates for n strips.
Doubling the first and last ordinates.
Only the interior ordinates are doubled.

Key takeaways

  • h = (b − a)/n; n strips need n+1 ordinates.
  • Area ≈ (h/2)[y₀ + yₙ + 2(interior)].
  • Convex curve → over-estimate; concave → under-estimate.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The strip width h equals:

    • b−an\frac{b-a}{n}
    • nb−a\frac{n}{b-a}
    • b−ab - a
    • b−a2\frac{b-a}{2}
    Show answer and explanation

    Answer: b−an\frac{b-a}{n}

    Range over strips.

  2. For n strips, the number of ordinates is:

    • n+1n + 1
    • nn
    • n−1n - 1
    • 2n2n
    Show answer and explanation

    Answer: n+1n + 1

    Endpoints included.

  3. In the rule, the interior ordinates are:

    • doubled
    • halved
    • unchanged
    • squared
    Show answer and explanation

    Answer: doubled

    Weight 2.

  4. For a convex curve the trapezium rule:

    • over-estimates
    • under-estimates
    • is exact
    • gives zero
    Show answer and explanation

    Answer: over-estimates

    Trapezia above curve.

  5. The first and last ordinates have weight:

    • 1
    • 2
    • 0
    • ½
    Show answer and explanation

    Answer: 1

    Only interior doubled.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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