The trapezium rule

The rule

∫_a^{b} y dx ≈ (h/2)(y_0 + 2(y_1 + y_2 + ... + y_n-1) + y_n), where h = (b - a)/n.

n strips need n + 1 ordinates (y-values). The first and last are counted once; all the others twice.

Over or under?

• Curve bending upwards (convex, like y = x²): each trapezium sits over the curve, so the rule overestimates.
• Curve bending downwards (concave, like y = sqrt(x)): the rule underestimates.

Better accuracy

Use more strips of smaller width. Percentage error = (estimate - exact)/exact x 100.

Worked example

Use the trapezium rule with 4 strips to estimate ∫₀^{2} x² dx.

1. h = (2 - 0)/4 = 0.5
2. y-values: 0, 0.25, 1, 2.25, 4
3. (0.5/2)(0 + 2(0.25 + 1 + 2.25) + 4) = 0.25 x 11

Answer: 2.75 (the exact value is 8/3, so this is an overestimate)

Check yourself

1. The trapezium rule is used with 4 strips to estimate ∫₁^{3} f(x) dx. Find h.
   Show answer

   0.5. h = (3 - 1)/4 = 0.5.

2. How many ordinates (y-values) do you need for the trapezium rule with 5 strips?
   Show answer

   6. n strips need n + 1 ordinates: 6.

3. Use the trapezium rule with 4 strips to estimate ∫₀^{1} sqrt(1 + x²) dx. Give your answer to 2 decimal places.
   Show answer

   1.15. h = 0.25. y-values: 1, 1.0308, 1.1180, 1.2500, 1.4142. Estimate = 0.125 x (1 + 1.4142 + 2(1.0308 + 1.1180 + 1.2500)) = 1.15.

4. Use the trapezium rule with 4 strips to estimate ∫₁^{3} (1/x) dx. Give your answer to 3 decimal places.
   Show answer

   1.117. h = 0.5. y-values: 1, 0.6667, 0.5, 0.4, 0.3333. Estimate = 0.25 x (1 + 0.3333 + 2(0.6667 + 0.5 + 0.4)) = 1.117.

5. The curve y = x² bends upwards between x = 0 and x = 2. What does the trapezium rule give for the area?
   • An underestimate
   • The exact value
   • An overestimate
   • It depends on the number of strips
   Show answer

   An overestimate. For a convex curve the straight tops of the trapeziums lie over the curve.

6. How can you make a trapezium rule estimate more accurate?
   • Use more strips of smaller width
   • Use fewer, wider strips
   • Count the first and last ordinates twice
   • Round the y-values to 1 decimal place
   Show answer

   Use more strips of smaller width. Narrower strips follow the curve more closely.

7. The trapezium rule with 4 strips gives 2.75 for ∫₀^{2} x² dx, whose exact value is 8/3. Find the percentage error to 1 decimal place.
   Show answer

   3.1 %. (2.75 - 8/3)/(8/3) x 100 = 3.1%.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.