What is the trapezoidal rule formula?

Divide the interval into n equal strips of width h. The estimate is h/2 times (first ordinate + last ordinate + twice the sum of all the others).

How do I know if the trapezoidal rule over-estimates?

Check the second derivative on the interval. If it is positive the curve is concave upwards, the chords lie above the curve, and the rule over-estimates. If negative, it under-estimates.

Is error estimation required?

No. The curriculum notes say error estimation for the trapezoidal rule is not required, and only equal-width subintervals are used.

The Trapezoidal Rule (DSE M1): Formula and Over- or Under-estimate

The trapezoidal rule with n strips of width h = (b − a)/n estimates ∫_a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + … + y_n−1) + y_n]. If the curve is concave upwards (f″ > 0) it is an over-estimate; if concave downwards, an under-estimate.

Solve The Trapezoidal Rule questions with AI

The trapezoidal rule on y = e^(x²) from 0 to 1 as the number of strips rises from 1 to 8. With 4 strips the estimate is 1.4907, as in Example 1. Try it yourself →

Trapezoidal Rule Formulas

Strip widthh = (b − a) / n

Rule(h/2)[y₀ + 2(y₁ + y₂ + … + y_n−1) + y_n]

Concave upwards (f″ > 0)Over-estimate

Concave downwards (f″ < 0)Under-estimate

Worked Examples — The Trapezoidal Rule

Example 1: Estimating an integral with 4 strips

Use the trapezoidal rule with 4 subintervals to estimate ∫₀¹ e^x² dx, and determine whether the estimate is an over-estimate or an under-estimate.

h = (1 − 0)/4 = 0.25
y₀ = 1, y₁ = e^0.0625 ≈ 1.0645, y₂ = e^0.25 ≈ 1.2840, y₃ = e^0.5625 ≈ 1.7551, y₄ = e ≈ 2.7183
(0.25/2)[1 + 2(1.0645 + 1.2840 + 1.7551) + 2.7183] ≈ 1.4907
d²y/dx² = (2 + 4x²)e^x² > 0, so the curve is concave upwards

Answer: ≈ 1.4907, an over-estimate

Example 2: Why more strips give a better estimate

Explain what happens to the trapezoidal estimate of ∫₀¹ e^x² dx when the number of strips is doubled.

Each trapezium has a straight top joining two points on the curve.
With narrower strips the straight tops lie closer to the curve.
The estimate falls from 1.4907 (4 strips) to about 1.4697 (8 strips), towards the true value of about 1.4627.

Answer: The estimate gets closer to the true value, while remaining an over-estimate.

💡 Exam Traps & Secrets

⚠️ n strips need n + 1 valuesFour subintervals use five ordinates, y₀ to y₄. Only the first and last are counted once; all the others are doubled.

⚠️ Justify with the second derivativeSaying "it looks concave upwards" is not enough. Find d²y/dx², state its sign on the interval, then conclude over- or under-estimate.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Trapezoidal rule)Use the trapezoidal rule with 5 subintervals to estimate the integral of 1/(1 + x²) from 0 to 1.

(Concave downwards)Estimate the integral of ln x from 1 to 3 with 4 strips and state whether it is an over- or under-estimate.

(Effect of n)Show the trapezoidal rule for y = e^(x²) on 0 to 1 with 2, 4 and 8 strips.