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AI HL · 13.5

Integration

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13.5.7

The trapezium rule

13.5.7

The trapezium rule

Target: I can write the trapezium rule and say what h, n and the y-values are. (+3 more)

By the end of this section

Key points

The rule

  • Many functions cannot be integrated exactly by hand. The trapezium rule gives an estimate of the area under the curve.
  • Cut the area into n strips of equal width. Each strip is close to a trapezium.
  • \(\int_a^b f(x)\,dx \approx \tfrac12 h\,[\,(y_0+y_n)+2(y_1+y_2+\dots+y_{n-1})\,]\), where \(h=\dfrac{b-a}{n}\).
  • Remember it as: both ends, plus twice the middle values.
  • One strip is a trapezium with parallel sides \(y_i\) and \(y_{i+1}\) and width h. Its area is \(\tfrac12 h(y_i+y_{i+1})\).

Using it

  • n is the number of strips. There are n + 1 y-values: do not use n as the number of values.
  • Build a table of x and y first. The rule is much easier to see.
  • Start with \(x_0=a\), then add h each time until you reach b.
  • Keep extra figures in the y-values and round only the final answer, to 3 significant figures.
  • More strips give a more accurate estimate.

Tables of values

  • If you are given a table of x and y values with equal gaps, h is the gap between x-values.
  • Apply the same rule to the y-values in the table. There is no function to evaluate.

On the calculator

Table of values

  • Enter the function as f(x) on a graphs page, then open the table of values. Or put the x-values in a list and the formula in the next column.
  • Read each y-value from the table. Do not round until the end of the working.
  • Type the whole bracket on a calculator page, or add the values from the table by hand.

Screenshot and video to come

The value of an integral

  • On a calculator page, use the definite integral template. Enter the function, x, and the two limits.
  • Use it to check a trapezium estimate: the two values should be close.
  • The trapezium estimate is only an approximation. The calculator value is the one to compare it with.

Screenshot and video to come

Explore: the trapezium rule

Choose a function and the limits a and b. Change the number of strips n and compare the estimate with the calculator value. Or add the strips one at a time.

Trapezium estimate
Calculator value
Percentage error
xyTimes used

Both ends once, the middle values twice. The working shows y to 3 decimal places; the estimate uses more. For sin x the calculator is in radians.

Change n to see the estimate move.

  1. Set n = 1, then n = 2, then n = 4. What happens to the estimate and to the percentage error? Why?
  2. For which functions is the estimate too big, and for which too small? What do those curves look like between the chords?
  3. Choose sin x and move b past 3. When the curve changes from concave down to concave up, what happens to the over- and under-estimate?
  4. In the table, why are the first and last y-values used once but the others twice?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A reciprocal function

Use the trapezium rule with 4 strips to estimate \(\int_{1}^{5} \frac{1}{x}\,dx\).

123450.20.40.60.811.2xy
Working
  1. Strip width

    \(h=\dfrac{5-1}{4}=1\)

    The interval is cut into 4 equal strips.

  2. Table of values

    x: 1, 2, 3, 4, 5; y: 1.000, 0.5000, 0.3333, 0.2500, 0.2000

    4 strips need 5 y-values. Keep 4 significant figures in the table.

  3. The rule

    \(\tfrac12(1)[(1.000+0.2000)+2(0.5000+0.3333+0.2500)]\) = 1.683

    Both ends once, the middle values twice, then ½h.

  4. Answer

    ≈ 1.68

    Give 3 significant figures.

Video walkthrough coming
A multiple of eˣ

Use the trapezium rule with 4 strips to estimate \(\int_{0}^{2} 4e^x\,dx\).

Working
  1. Strip width

    \(h=\dfrac{2-0}{4}=0.5\)

    The interval is cut into 4 equal strips.

  2. Table of values

    x: 0, 0.5, 1, 1.5, 2; y: 4.000, 6.595, 10.87, 17.93, 29.56

    4 strips need 5 y-values. Keep 4 significant figures in the table.

  3. The rule

    \(\tfrac12(0.5)[(4.000+29.56)+2(6.595+10.87+17.93)]\) = 26.09

    Both ends once, the middle values twice, then ½h.

  4. Answer

    ≈ 26.1

    Give 3 significant figures.

Video walkthrough coming
Natural logarithm

Use the trapezium rule with 4 strips to estimate \(\int_{3}^{7} \ln x\,dx\).

Working
  1. Strip width

    \(h=\dfrac{7-3}{4}=1\)

    The interval is cut into 4 equal strips.

  2. Table of values

    x: 3, 4, 5, 6, 7; y: 1.099, 1.386, 1.609, 1.792, 1.946

    4 strips need 5 y-values. Keep 4 significant figures in the table.

  3. The rule

    \(\tfrac12(1)[(1.099+1.946)+2(1.386+1.609+1.792)]\) = 6.310

    Both ends once, the middle values twice, then ½h.

  4. Answer

    ≈ 6.31

    Give 3 significant figures.

Video walkthrough coming
A logarithm with a multiple of x

Use the trapezium rule with 5 strips to estimate \(\int_{2}^{3} \ln(5x)\,dx\).

Working
  1. Strip width

    \(h=\dfrac{3-2}{5}=0.2\)

    The interval is cut into 5 equal strips.

  2. Table of values

    x: 2, 2.2, 2.4, 2.6, 2.8, 3; y: 2.303, 2.398, 2.485, 2.565, 2.639, 2.708

    5 strips need 6 y-values. Keep 4 significant figures in the table.

  3. The rule

    \(\tfrac12(0.2)[(2.303+2.708)+2(2.398+2.485+2.565+2.639)]\) = 2.518

    Both ends once, the middle values twice, then ½h.

  4. Answer

    ≈ 2.52

    Give 3 significant figures. Put 5x into the logarithm for each x-value: the y-value is ln(5x), not 5 ln x.

Video walkthrough coming
From a table

A sprinter's speed v (m/s) is measured every 2 seconds. The area under the speed–time graph is the distance run. x: 0, 2, 4, 6, 8 y: 0, 5.8, 8.1, 9.2, 9.6 Use the trapezium rule to estimate the distance run in the first 8 seconds.

Working
  1. Strip width

    h = 2

    The x-values are 2 apart, so each strip is 2 wide. There are 4 strips.

  2. The rule

    \(\tfrac12(2)[(0+9.600)+2(5.800+8.100+9.200)]\) = 55.80

    Both ends once, the middle values twice. The y-values are given, so there is no table to build.

  3. Answer

    ≈ 55.8

    The area is a distance, in metres.

Video walkthrough coming