Library / Foundations: Maths / FMA-28

Foundations: Maths

Trapezium rule (4 strips)

Numerical estimate of the area under a curve from five ordinates.

Ref: FMA-28
Rev: v1
Field: Foundations: Maths
Result: A

\int y dx \approx \frac{h}{2} [y_{0} + 2 (y_{1} + y_{2} + y_{3}) + y_{4}]

h = (b − a)/n is the strip width. The two end ordinates count once and the interior ordinates twice. The calculator uses n = 4 strips.

Split the area into strips, treat each as a trapezium and add them: the ordinates at the ends count once and those inside count twice.

Calculator

Solve for

AArea estimateComputed

hStrip width h

0.12

y₀Ordinate y₀

0.55

y₁Ordinate y₁

0.55

y₂Ordinate y₂

0.55

y₃Ordinate y₃

0.56

y₄Ordinate y₄

Interval covered 4h2
Mean ordinate2.375

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Area estimate () current value 4.75

hStrip width hKey driver

0.9519

y₃Ordinate y₃

3.386.13

y₁Ordinate y₁

4.386.63

y₂Ordinate y₂

46.25

y₄Ordinate y₄

3.755.5

y₀Ordinate y₀

4.635.75

For a curve that bends upwards (convex) the rule overestimates; for one that bends downwards it underestimates.
More strips give a better estimate; doubling the strips cuts the error by about four.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
h Strip width h	0.1 to 2	0.95	19	18.05	+10%	+10 %
y₃ Ordinate y₃	0.5 to 6	3.375	6.125	2.75	+10%	+3.42 %
y₁ Ordinate y₁	0.5 to 5	4.375	6.625	2.25	+10%	+1.32 %
y₂ Ordinate y₂	0.5 to 5	4	6.25	2.25	+10%	+2.11 %
y₄ Ordinate y₄	1 to 8	3.75	5.5	1.75	+10%	+2.63 %
y₀ Ordinate y₀	0.5 to 5	4.625	5.75	1.125	+10%	+0.526 %

Sweep of Strip width h

Strip width h	Area estimate
0.1	0.95
0.1454	1.382
0.2115	2.009
0.3075	2.922
0.4472	4.249
0.6503	6.178
0.9457	8.985
1.375	13.07
2	19

Sweep one variable

Vary

Area estimate ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Area estimate

Result ·

What the equation gives you. Everything else feeds it.

Strip width h

· typical 0.1 to 2

Width of each strip. Directly proportional.

y₀

Ordinate y₀

· typical 0.5 to 5

First ordinate. Counts once.

y₁

Ordinate y₁

· typical 0.5 to 5

Second ordinate. Counts twice.

y₂

Ordinate y₂

· typical 0.5 to 5

Third ordinate. Counts twice.

y₃

Ordinate y₃

· typical 0.5 to 6

Fourth ordinate. Counts twice.

y₄

Ordinate y₄

· typical 1 to 8

Last ordinate. Counts once.

Worked example with the values above

Ends: y₀ + y₄ = 6. Interior twice: 2 × 6.5 = 13.
Sum in brackets: 19.
A ≈ (h/2) × sum = 4.75.

Assumptions and limits

Equal strip widths.
The curve is smooth between ordinates.

Common mistakes

Counting the end ordinates twice.
Using the wrong strip width.

Statistics and uncertainty

The error scales with h², so halving the strip width reduces the error roughly fourfold; ordinates read from a graph add their own reading error.

Where it is used

Area under a measured speed–time or force–distance graph.
Energy from sampled power data.

Maths you need

Area of a trapezium: ½(a + b)h
Integration: area under a curve

Related formulas

Practise this formula