NCEA Level 3Trapezium and Simpson's rulesAS 91579

Trapezium and Simpson's rules: numerical areas

What this lesson covers

  • When a function cannot be integrated, or only data is given, the area is estimated from ordinates.
  • The strip width h is the interval width divided by the number of strips, not the number of ordinates.
Key idea

When you cannot integrate

  • No antiderivative, or only data.
  • Estimate from the heights: ordinates.
  • Equally spaced values of x.
Key idea

Both rules are in the booklet

  • Trapezium rule and Simpson's rule.
  • Know what goes into them.
  • h = (b − a) ÷ number of strips.
Worked example

y = 1/x, from 1 to 3, four strips

  • Each strip is a trapezium.
  • h = (3 − 1)/4 = 0.5.
  • The tops sit just above the curve.
y = 1/x, from 1 to 3, four strips
Worked example

Four strips, five ordinates

  • One ordinate at each strip edge.
  • Ends count once.
  • Middle ordinates count twice.
xy = 1/xWeight
111
1.52/32
21/22
2.52/52
31/31
Worked example

The trapezium rule

  • Ends once, middle twice.
  • Exact fractions, then 4 d.p.
  1. A ≈ (h/2)[ends + 2 × middle]
  2. h/2 = 0.5/2 = 1/4
  3. ends: 1 + 1/3 = 4/3
  4. middle: 2/3 + 1/2 + 2/5
  5. = 20/30 + 15/30 + 12/30 = 47/30
  6. 2 × 47/30 = 47/15
  7. bracket = 20/15 + 47/15 = 67/15
  8. A ≈ (1/4)(67/15) = 67/60
  9. ≈ 1.1167
Worked example

The wrong h: a 2025 error

  • Strips, not ordinates.
  • Strips = ordinates − 1.
  1. bracket = 67/15
  2. h = (3 − 1)/5 = 0.45 ordinates
  3. 0.2 × 67/15 = 67/75 ≈ 0.8933
  4. h = (3 − 1)/4 = 0.54 strips
  5. 0.25 × 67/15 = 67/60 ≈ 1.1167
Key idea

Simpson's rule

  • An even number of strips.
  • Weights 1, 4, 2, 4, …, 4, 1.
  • Multiply by h/3, not h/2.
Worked example

Simpson's rule on y = 1/x

  • Weights: odd 4, even 2.
  • Then multiply by h/3.
  1. A ≈ (h/3)[ends + 4 odd + 2 even]
  2. h/3 = 0.5/3 = 1/6
  3. ends: 1 + 1/3 = 4/3
  4. 4(2/3 + 2/5) = 4(16/15) = 64/15
  5. 2(1/2) = 1the even ordinate
  6. bracket = 4/3 + 64/15 + 1
  7. = 20/15 + 64/15 + 15/15 = 99/15
  8. A ≈ (1/6)(99/15) = 99/90
  9. = 11/10 = 1.1
Worked example

How close are they?

  • Exact: ln 3 ≈ 1.0986.
  • Trapezium is over: the curve bends up.
  • Simpson's is much closer.
Value
exactln 3 ≈ 1.0986
trapezium≈ 1.1167, over
Simpson's1.1000
Worked example

From a table: a car speeding up

  • Distance = area under the speed graph.
  • 7 readings, so 6 strips.
  • 6 is even, so both rules work.
t (s)v (m/s)Simpson
001
254
492
6124
8142
10154
12151
Worked example

Distance: the trapezium rule

  • h = 2, so h/2 = 1.
  • Ends once, middle twice.
  1. h = (12 − 0)/6 = 2, h/2 = 1
  2. ends: 0 + 15 = 15
  3. middle: 5 + 9 + 12 + 14 + 15 = 55
  4. d ≈ 1 × [15 + 2(55)]trapezium rule
  5. = 15 + 110 = 125 m
Worked example

Distance: Simpson's rule

  • Simpson's: h/3 = 2/3.
  • Units, and what it means.
  1. h/3 = 2/3
  2. ends: 0 + 15 = 15
  3. 4(5 + 12 + 15) = 4(32) = 128t = 2, 6, 10
  4. 2(9 + 14) = 2(23) = 46t = 4, 8
  5. bracket = 15 + 128 + 46 = 189
  6. d ≈ (2/3)(189) = 126 m
  7. about 126 m in the first 12 s
Key idea

Show every ordinate

  • Write the rule with each ordinate.
  • Full accuracy, then 3 s.f.
  • Odd strips: no Simpson's.
Key idea

Where the marks are

  • Achieved: the trapezium rule, right h.
  • Merit, Excellence: part of a bigger problem.
  • Compare with exact, or use in context.
Summary

What to take away

  • h is the interval width over the number of strips.
h(b − a) ÷ strips
Ordinatesstrips + 1
Trapeziumends + 2 × middle
Simpson'sends + 4 odd + 2 even
Simpson's nmust be even
Show itevery ordinate
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 3 uses the integral in 2.
  • Write out h and every ordinate.
GivenFind
1x = 1 to 5, 8 stripsh, and ordinates
2∫₀⁴ √x dx, 4 stripstrapezium rule
3the same integralSimpson's rule
4y: 3, 5, 6, 4, 2Simpson's, x = 0 to 4
Show answer

1. h = (5 − 1)/8 = 0.5, with 9 ordinates. 2. h = 1, ordinates 0, 1, √2, √3, 2: 0.5[0 + 2 + 2(1 + √2 + √3)] ≈ 5.15. 3. (1/3)[0 + 2 + 4(1 + √3) + 2√2] ≈ 5.25, closer to the exact 16/3 ≈ 5.33. 4. (1/3)[3 + 2 + 4(5 + 4) + 2(6)] = 53/3 ≈ 17.7.

Worked example

The answers: questions 1 and 2

  1. 1. h = (5 − 1)/8 = 0.5
  2. 8 strips, so 9 ordinates
  3. 2. h = (4 − 0)/4 = 1
  4. ordinates: 0, 1, √2, √3, 2
  5. (1/2)[0 + 2 + 2(1 + √2 + √3)]trapezium rule
  6. = (1/2)(4 + 2√2 + 2√3)
  7. = (1/2)(10.2925) = 5.14634 d.p.
  8. ≈ 5.15
Worked example

The answers: questions 3 and 4

  1. 3. (1/3)[0 + 2 + 4(1 + √3) + 2√2]
  2. = (1/3)(6 + 4√3 + 2√2)
  3. = (1/3)(15.7566) = 5.25224 d.p.
  4. ≈ 5.25
  5. exact: (2/3)(8) = 16/3 ≈ 5.3333Simpson is closer
  6. 4. (1/3)[3 + 2 + 4(5 + 4) + 2(6)]
  7. = (1/3)(5 + 36 + 12) = 53/3
  8. ≈ 17.7

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