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IBMaths: AI HLDPRevision Notes5. Calculus5.3 Integration5.3.1 Trapezoid Rule: Numerical Integration

Trapezoid Rule: Numerical Integration (DP IB Maths: AI HL)

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Trapezoid Rule: Numerical Integration

What is the trapezoid rule?

  • The trapezoidal rule is a numerical method used to find the approximate area enclosed by a curve, thespace x-axis and two vertical lines
    • it is also known as ‘trapezoid rule’ and ‘trapezium rule’

  • The trapezoidal rule finds an approximation of the area by summing of the areas of trapezoids beneath the curve
    • y subscript 0 equals f left parenthesis a right parenthesis comma space space space y subscript 1 equals f left parenthesis a plus h right parenthesis comma space space space y subscript 2 equals f left parenthesis a plus 2 h right parenthesis etc
integral subscript a superscript b f left parenthesis x right parenthesis space d x almost equal to 1 half h stretchy left square bracket stretchy left parenthesis y subscript 0 plus y subscript n stretchy right parenthesis plus 2 stretchy left parenthesis y subscript 1 plus y subscript 2 plus... plus y subscript n minus 1 end subscript stretchy right parenthesis stretchy right square bracket space
where h equals fraction numerator b minus a over denominator n end fraction
  • Note that there are n trapezoids (also called strips) but left parenthesis n plus 1 right parenthesis function values left parenthesis y subscript i right parenthesis

  • The trapezoidal rule is given in the formula booklet

What else can I be asked to do with the trapezoid rule?

  • Comparing the true answer with the answer from the trapezoid rule
    • This may involve finding the percentage error in the approximation
    • The true answer may be given in the question, found from a GDC or from work on integration

Examiner Tip

  • Ensure you are clear about the difference between the number of data points (y values) and the number of strips (number of trapezoids) used in a Trapezoid Rule question
  • Although it shouldn't be too much trouble to type the trapezoid rule into your GDC in one go, it may be wise to work parts of it out separately and write these down as part of your working out

Worked example

a)
Using the trapezoidal rule, find an approximate value for
integral subscript 0 superscript 4 fraction numerator 6 x squared over denominator x cubed plus 2 end fraction d x
to 3 decimal places, using n equals 4.

5-2-1-ib-si-ai-only-trap-rule-we-old-crop-a

b)
Given that the area bounded by the curve , the x-axis and the lines x equals 0 and x equals 4 is 6.993 to three decimal places, calculate the percentage error in the trapezoidal rule approximation.

5-2-1-ib-si-ai-only-trap-rule-we-old-crop-b

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.