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The Trapezium Rule

Q: Why are the interior y-values doubled but not the first and last?

Each interior boundary is shared between two adjacent trapeziums, so its height contributes to both areas — effectively counting it twice in the combined formula. The very first and last boundaries only belong to one trapezium each.

Q: How do I know if my estimate is too high or too low?

Check the concavity of the curve: if it curves upward (convex, second derivative positive), the straight trapezium tops lie above the curve, giving an overestimate. If it curves downward (concave), they lie below, giving an underestimate.

Q: How can I improve the accuracy of my estimate?

Use more (and therefore thinner) strips — as the number of strips increases, each trapezium fits the curve more closely and the total error shrinks.

Some functions, however important, simply cannot be integrated exactly using any technique from this course. Rather than giving up, split the area under the curve into a series of trapeziums instead of exact slices — and add up their areas for a very good estimate.

30 min Video by TLMaths Numerical Methods

The big picture

The $trapezium rule$ is a numerical method for estimating a definite integral without needing to find an antiderivative at all — essential for functions with no elementary antiderivative. It works by approximating the smooth curve with a series of straight-line segments, turning the area under the curve into a sum of trapezium areas, which get more accurate as you use more (thinner) strips.

A-Level Maths I3-03: Numerical Integration: Introducing the Trapezium RuleWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Understand the trapezium rule as an approximation using straight-line strips
Apply the trapezium rule formula to estimate a definite integral
Recognise whether the trapezium rule over- or under-estimates the true area
Increase accuracy by using more strips

The trapezium rule formula

Divide the interval $[a, b]$ into $n$ equal strips of width $h = \frac{b - a}{n}$ . At each strip boundary, evaluate $y = f (x)$ . Each strip is approximated as a trapezium (rather than the exact curved region), and their areas are summed using a formula that only double-counts the interior boundary heights once.

\int_{a}^{b} y d x \approx \frac{h}{2} [y_{0} + y_{n} + 2 (y_{1} + y_{2} + \dots + y_{n - 1})]

First + last height, plus double every interior height, times half the strip width.

1Strip width:

h = \frac{4 - 0}{4} = 1

2Compute the y-values at

x = 0, 1, 2, 3, 4

y_{0} = 0

y_{1} = 1

y_{2} = 4

y_{3} = 9

y_{4} = 16

3Apply the formula:

\frac{1}{2} [0 + 16 + 2 (1 + 4 + 9)] = \frac{1}{2} [16 + 2 (14)] = \frac{1}{2} (44)

Answer

22

(the exact value is

\int_{0}^{4} x^{2} d x = \frac{64}{3} \approx 21.33

, so this is a close overestimate)

Tip — Always set up a clear table of $x$ and $y$ values first — it makes identifying which $y$ -values are "first/last" (counted once) versus "interior" (counted twice) far less error-prone.

Over- and under-estimates

Whether the trapezium rule over- or under-estimates the true area depends on the curve's concavity. If the curve is $convex$ (curving upward, like $x^{2}$ ), the straight-line trapezium tops sit ABOVE the curve, giving an OVERESTIMATE. If the curve is $concave$ (curving downward), the trapeziums sit BELOW the curve, giving an UNDERESTIMATE.

y = x^{2}

curves upward (convex) throughout the interval.

2A straight line joining two points on a convex curve lies ABOVE the curve between them.

3So the trapezium rule overestimates the true area.

AnswerIt overestimates — confirmed by the earlier result (22 vs the true value 21.33).

Why this works

This connects directly to the second derivative: $\frac{d ^{2} y}{d x ^{2}} > 0$ (convex) gives an overestimate; $\frac{d ^{2} y}{d x ^{2}} < 0$ (concave) gives an underestimate.

Improving accuracy

Using MORE strips (a larger $n$ , so a smaller $h$ ) makes each individual trapezium a closer fit to the curve, reducing the overall error — the trapezium rule estimate approaches the true integral as $n \to \infty$ .

1More (thinner) strips means each trapezium spans a smaller section of the curve.

2Over a smaller section, the curve is closer to a straight line, so the trapezium approximation is more accurate.

AnswerThinner strips fit the curve more closely, reducing the total approximation error.

Tip — Comparing estimates from two different numbers of strips (e.g. doubling $n$ ) is a common exam technique for showing the trapezium rule is converging toward the true value.

Think like an examiner

Set out a clear table of $x$ and $y$ values before applying the formula — this is usually worth method marks on its own.
State clearly whether your answer is an over- or underestimate, and justify it using the shape (convexity/concavity) of the curve.
Double-check you have correctly identified which $y$ -values are interior (doubled) versus the first/last (not doubled).

Common misconceptions

Doubling every y-value, including the first and last.

Only the INTERIOR y-values (

y_{1}

through

y_{n - 1}

) are doubled — the first (

y_{0}

) and last (

y_{n}

) are counted once each.

Using the wrong strip width, e.g. forgetting to divide

(b - a)

n

Strip width

h = \frac{b - a}{n}

— always calculate this first before finding the y-values at each boundary.

Guessing whether the estimate is an over- or underestimate without justification.

Refer to the curve's concavity (convex → overestimate; concave → underestimate) as your reasoning.

Trapezium rule

\int_{a}^{b} y d x \approx \frac{h}{2} [y_{0} + y_{n} + 2 (y_{1} + \dots + y_{n - 1})]

Trapezium rule with strip width h = (b−a)/n.

Stretch yourself

Use the trapezium rule with 5 strips to estimate $\int_{0}^{5} x + 1 d x$ , giving your answer to 3 decimal places.

Hint — Strip width is 1. Compute y-values at x = 0, 1, 2, 3, 4, 5, then apply the trapezium rule formula.

Questions students ask

Why are the interior y-values doubled but not the first and last?

Each interior boundary is shared between two adjacent trapeziums, so its height contributes to both areas — effectively counting it twice in the combined formula. The very first and last boundaries only belong to one trapezium each.

How do I know if my estimate is too high or too low?

Check the concavity of the curve: if it curves upward (convex, second derivative positive), the straight trapezium tops lie above the curve, giving an overestimate. If it curves downward (concave), they lie below, giving an underestimate.

How can I improve the accuracy of my estimate?

Use more (and therefore thinner) strips — as the number of strips increases, each trapezium fits the curve more closely and the total error shrinks.

Key takeaways

The trapezium rule estimates $\int_{a}^{b} y d x \approx \frac{h}{2} [y_{0} + y_{n} + 2 (y_{1} + \dots + y_{n - 1})]$ , with $h = \frac{b - a}{n}$ .
A convex curve gives an overestimate; a concave curve gives an underestimate.
More strips (smaller h) generally improve accuracy.
Always tabulate x and y values clearly before applying the formula.

How this fits the course

Build on

Vectors in Two Dimensions

Leads to

Vectors in Two Dimensions

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The Trapezium Rule

30 min Video by TLMaths Numerical Methods