Radford Mathematics IB Mathematics resources • AI SL

IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus

Trapezoidal Rule for Integration

Approximate area from equally spaced data or function values with the trapezoidal rule.

AI SL · SL 5.8

Learning goal

Estimate areas using the trapezoidal rule with equal-width intervals.

Syllabus link

AI SL: SL 5.8 · Topic 5 Calculus.

Big idea

The trapezoidal rule replaces the curved boundary by straight-line segments and adds the resulting trapezia.

Key relationship

\(A\approx\frac{h}{2}\left[y_0+2(y_1+\cdots+y_{n-1})+y_n\right]\)

1

Core idea

The trapezoidal rule replaces the curved boundary by straight-line segments and adds the resulting trapezia.

Estimate areas using the trapezoidal rule with equal-width intervals.

2

Key results and method

\[A\approx\frac{h}{2}\left[y_0+2(y_1+\cdots+y_{n-1})+y_n\right]\]
\[h=\frac{b-a}{n}\]
  1. Check that the x-values are equally spaced.
  2. Identify the interval width \(h\).
  3. Use endpoint ordinates once and interior ordinates twice.
  4. Include square units or contextual units.
3

Worked example

Question

Use \(x=0,1,2,3\) with \(y=1,2,4,5\) to estimate the area.

Solution
  1. Here \(h=1\).
  2. \(A\approx\tfrac12[1+2(2+4)+5]\).
Answer: \(A\approx9\) square units.
4

Practice

Try these questions before moving on.

  1. Apply the rule to a table with five ordinates.
  2. Explain how the estimate changes if more, narrower intervals are used.
  3. Decide whether a displayed concave-up curve is likely to be over- or underestimated.

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5

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