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Trapezium Rule (Strip)

Approximating area under a curve.

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A \approx \frac{h}{2} (y_{0} + y_{1})

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Core idea

Overview

The Trapezium Rule provides a linear approximation for the area under a curve by treating a single interval as a trapezoid. It essentially calculates the product of the interval width and the arithmetic mean of the two boundary heights.

When to use: Use this method when you need to estimate a definite integral for data sets where only discrete points are known. It is most effective when the interval width is small or the function behaves relatively linearly across the chosen segment.

Why it matters: This rule is fundamental in numerical analysis and engineering for calculating total values from rate data, such as finding distance from velocity or energy from power. It provides a simple yet effective way to process sensor data in real-time computing environments.

Symbols

Variables

A = Strip Area, h = Width, y_0 = Left Height, y_1 = Right Height

A

Strip Area

u ni t s^{2}

h

Width

V a r iab l e

y_{0}

Left Height

V a r iab l e

y_{1}

Right Height

V a r iab l e

Walkthrough

Derivation

Derivation of the Trapezium Rule Formula

The trapezium rule approximates area under a curve by splitting the interval into trapezia and summing their areas.

Interval [a,b] is split into n equal strips of width h.
Ordinates are $y_{0}, y_{1}, \dots, y_{n}$ .

Area of One Trapezium:

Each strip forms a trapezium with parallel sides $y_{0}$ and $y_{1}$ .

A_{1} = \frac{1}{2} h (y_{0} + y_{1})

Add Areas of All Strips:

Sum the trapezium areas across the interval.

Area \approx \frac{1}{2} h (y_{0} + y_{1}) + \frac{1}{2} h (y_{1} + y_{2}) + \dots + \frac{1}{2} h (y_{n - 1} + y_{n})

Collect Like Terms:

Interior ordinates are counted twice because each is shared by two trapezia.

Area \approx \frac{1}{2} h [y_{0} + y_{n} + 2 (y_{1} + y_{2} + \dots + y_{n - 1})]

Result

Area \approx \frac{1}{2} h [y_{0} + y_{n} + 2 (y_{1} + y_{2} + \dots + y_{n - 1})]

Source: Edexcel A-Level Mathematics — Pure (Numerical Methods)

Visual intuition

Graph

The graph is a linear function where the strip area increases proportionally with the height of the ordinates. Since the formula represents a constant width multiplied by the average of two values, the plot of area against the sum of the ordinates forms a straight line passing through the origin.

Graph type: linear

Why it behaves this way

Intuition

Imagine slicing the area under a curve into narrow vertical strips, then replacing the curved top of each strip with a straight line connecting the function values at its boundaries, forming a simple trapezoid.

The approximated area under the curve within a single interval.

This is the desired total quantity being estimated by the rule.

The width of the interval or 'strip' along the x-axis.

It defines how wide the segment is that we are approximating.

y_{0}

The function value (height) at the left boundary of the interval.

One of the two heights defining the parallel sides of the trapezoid.

y_{1}

The function value (height) at the right boundary of the interval.

The other height defining the parallel sides of the trapezoid.

(y_{0} + y_{1}) /2

The arithmetic mean of the function values at the interval boundaries.

Represents the average height of the trapezoid, which, when multiplied by the width 'h', gives its area.

Signs and relationships

/2: The division by 2 is a fundamental part of the area formula for a trapezoid, representing the averaging of the two parallel sides (y0 and y1) before multiplying by the height (h).
+: The addition combines the lengths of the two parallel sides (y0 and y1) to facilitate their averaging in the trapezoid area formula.

Free study cues

Insight

Canonical usage

The units of the approximated area are determined by the product of the units of the interval width and the function values, ensuring dimensional consistency.

Common confusion

A common mistake is incorrectly assuming the resulting area will always have units of length squared, rather than recognizing that the units are derived from the specific physical quantities represented by the

Unit systems

$h$ unit of independent variable (e.g., m, s) · Represents the width of the interval along the independent axis. Its unit defines one dimension of the area.

$y_{0}$ unit of dependent variable (e.g., m, m/s) · Represents the function value (height) at the start of the interval. Its unit defines the other dimension of the area.

$y_{1}$ unit of dependent variable (e.g., m, m/s) · Represents the function value (height) at the end of the interval. Must have the same unit as y_0.

$A$ product of units of h and y (e.g., m^2, m) · The approximated area will have units consistent with the product of the independent variable's unit and the dependent variable's unit.

One free problem

Practice Problem

A surveyor measures the depth of a river at two points 4 meters apart. The depth at the first point is 1.5 meters and at the second point is 2.2 meters. Estimate the cross-sectional area of this strip of the river.

Width4

Left Height1.5

Right Height2.2

Solve for: $A$

Hint: Multiply half the width by the sum of the two depths.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Area of a lake from map measurements.

Study smarter

Tips

Ensure the width h is the horizontal distance between y0 and y1.
Check if the curve is concave; this determines if your result is an over or under-estimate.
The result is exact for linear functions (first-degree polynomials).

Avoid these traps

Common Mistakes

Wrong h measurement.
Counting endpoints twice.

Common questions

Frequently Asked Questions

The trapezium rule approximates area under a curve by splitting the interval into trapezia and summing their areas.

Use this method when you need to estimate a definite integral for data sets where only discrete points are known. It is most effective when the interval width is small or the function behaves relatively linearly across the chosen segment.

This rule is fundamental in numerical analysis and engineering for calculating total values from rate data, such as finding distance from velocity or energy from power. It provides a simple yet effective way to process sensor data in real-time computing environments.

Wrong h measurement. Counting endpoints twice.