Year 12 Maths Advanced (2027) Integral calculus

The Trapezoidal Rule

20 practice questions 0 video lessons Theory + worked examples

Learn the trapezoidal rule for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). When a function has no easy primitive, or you only have a table of measurements, you can estimate an integral by joining equally spaced points with straight lines and adding the trapezium areas.

You will learn to choose the strip width, list the function values, apply the trapezoidal rule, and estimate areas from a table of measurements such as river depths or speeds — a practical numerical integration method used by surveyors and engineers.

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Practice questions

Every question with a fully worked solution.

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Theory

The trapezoidal rule estimates a definite integral using trapezoids. This Year 12 Mathematics Advanced topic (MAV-12-05) approximates areas when exact integration is hard or only data is given.

With \(n\) strips of width \(h=\dfrac{b-a}{n}\): \(\displaystyle\int_a^b f\,dx\approx\dfrac{h}{2}\big[y_0+2(y_1+\cdots+y_{n-1})+y_n\big]\).

The two end heights count once; every interior height counts twice. A single trapezoid is \(\dfrac{h}{2}(y_0+y_1)\).

Two trapezoids estimate the area under \(y=x^{2}\) on \([0,2]\).

\[\int_a^b f\,dx\approx\dfrac{h}{2}\big[y_0+2(y_1+\cdots+y_{n-1})+y_n\big],\ h=\dfrac{b-a}{n}\]

trapezoidal rule: h over 2 times y nought plus twice the interior values plus y n

Method

Find \(h=\dfrac{b-a}{n}\) and the heights \(y_0,\dots,y_n\).
Double the interior heights; add the two ends once.
Multiply the total by \(\dfrac{h}{2}\).

Example 1 — Two strips

Estimate \(\displaystyle\int_0^2 x^{2}\,dx\) with \(n=2\) (\(h=1\)).

Solution

\(\ \)	\(\approx\)	\(\tfrac12[0+2(1)+4]\)
	\(=\)	\(3\)

Example 2 — One trapezoid

Estimate \(\displaystyle\int_1^3 f\,dx\) with \(f(1)=2,\ f(3)=8\).

Solution

\(\ \)

\(\approx\)

\(\tfrac22[2+8]=10\)

Example 3 — From a table

Estimate \(\displaystyle\int_0^4 f\,dx\), \(h=2\), \(f(0)=1,f(2)=3,f(4)=5\).

Solution

\(\ \)

\(\approx\)

\(\tfrac22[1+2(3)+5]=12\)

Example 4 — Compare

The estimate for \(\displaystyle\int_0^2 x^{2}dx\) was \(3\). How does it compare?

Solution

\(\text{exact}\)	\(=\)	\(\tfrac83\approx2.67\)
\(\ \)		\(3>\tfrac83\ (\text{over-estimate})\)

Common pitfalls

Double the interior heights only. Ends count once.

Find \(h\) first: \(h=\dfrac{b-a}{n}\).

It is an estimate — upward-bending curves are over-estimated.

Frequently asked questions

What is the trapezoidal rule?

It estimates a definite integral by replacing the area with trapezoids: h over 2 times the first height, plus twice the interior heights, plus the last height.

When do you use the trapezoidal rule?

When a function is hard or impossible to integrate exactly, or when you only have a table of measured values.

Why are the interior values multiplied by 2?

Each interior height is shared by two adjacent trapezoids, so it is counted twice in the formula.

Does the trapezoidal rule over- or under-estimate?

For a curve that bends upward (concave up) it slightly over-estimates the true area.

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