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MathematicsGCSE Higher & A Level

What is trapezium rule?

The trapezium rule estimates the area under a curve by dividing it into strips of equal width and treating each strip as a trapezium.

Trapezium rule explained

You need the strip width hh, found from the range divided by the number of strips, and the y-values at every strip boundary. The rule doubles every interior ordinate because it appears in two trapezia.
Accuracy improves as the number of strips increases, so a common exam follow-up asks how to make the estimate better — the answer is more strips, not smaller limits.

Key formula

Area≈h2[y0+yn+2(y1+y2+⋯+yn−1)]\text{Area} \approx \dfrac{h}{2}\left[y_{0} + y_{n} + 2\left(y_{1} + y_{2} + \cdots + y_{n-1}\right)\right]

Worked example

With 44 strips of width 0.50.5 and ordinates 1,1.5,2.5,4,61, 1.5, 2.5, 4, 6: area ≈0.52[1+6+2(1.5+2.5+4)]=5.75\approx \frac{0.5}{2}\left[1 + 6 + 2(1.5 + 2.5 + 4)\right] = 5.75.

Examiner tip

Count ordinates, not strips: nn strips need n+1n + 1 y-values. Tabulating them keeps you from doubling the first or last by mistake.

Trapezium rule: common questions

How many strips should I use?
Use exactly the number stated in the question. Marks are awarded for the correct value of h and the correct ordinates, not for extra precision.
Why is the trapezium rule only an estimate?
It replaces each curved section with a straight chord, so the small regions between chord and curve are ignored.

Related terms

Area under a curve

The area under a curve is the region between the graph and the x-axis, found exactly by definite integration or estimated using the trapezium rule.

Definite integral

A definite integral evaluates an integral between two limits and returns a single numerical value, usually representing signed area.

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