Numerical area approximation and the integral idea | Year 12

01 / LEARN

The idea, step by step

Build area from narrow strips

When a curve has no convenient antiderivative or only sampled data are known, split its interval into strips. A rectangle uses one function value for its height. A trapezium uses the mean of the two endpoint heights. Multiply each height or mean height by the strip width.

For equal widths $h$ , add the strip areas. Include units: a velocity-time area has distance units, not speed units. With only sample values, the trapezium result assumes a straight segment between adjacent samples.

T = \frac{h}{2} (y_{0} + 2 y_{1} + \dots + 2 y_{n - 1} + y_{n})

Interpret the approximation

For an increasing positive function, left rectangles underestimate and right rectangles overestimate. Trapeziums need a curvature argument: for a convex curve they lie above the curve, but monotonicity alone does not decide their error.

Smaller strips often improve an approximation for a smooth function, but sample data and measurement error still matter. Do not report an approximate estimate as an exact measured total, and do not invent intermediate data.

Speed recorded at equally spaced times.
Time (s)	0	1	2	3
Speed (m/s)	0	2	3	6

A common trap

Adding the sampled heights without multiplying by interval width.

Instead: Identify the spacing first. Interior heights count twice in the trapezium formula; endpoints count once.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

For $f (x) = x^{2}$ on $[0, 2]$ , use two trapeziums of width 1.

The endpoint heights are 0,1,4.
$T = (0 + 1) / 2 + (1 + 4) / 2 = 3$ .
The exact integral is $8 / 3$ , so this is an overestimate.

T = 3

Example 2

Worked example

Use the supplied speed table to estimate distance by the trapezium rule.

Speed recorded at equally spaced times.
Time (s)	0	1	2	3
Speed (m/s)	0	2	3	6

$h = 1$ s. The three areas are 1,2.5 and 4.5 m.
Their sum is 8 m.
This estimates the integral from the samples; it is not exact unless the straight-segment model is justified.

8 m

Check your understanding

Use two left rectangles for $y = x^{2}$ on $[0, 2]$ .

Show answer

$1$

Heights 0 and 1, width 1: $1 (0 + 1) = 1$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

For $y = x^{2}$ on $[0, 2]$ , use two right rectangles.
3 marks
For $y = x^{2}$ on $[0, 2]$ , use midpoint rectangles of width 1.
3 marks
For $y = x^{2}$ on $[0, 2]$ , use four trapeziums.
3 marks
A rate is 4,6 and 10 L/min at 0,2 and 4 min. Estimate total volume by trapeziums.
3 marks
The function $f (x) = 2 x + 1$ is sampled at 0,1 and 2. Is the trapezium estimate exact? Explain.
3 marks
Why does the trapezium rule overestimate $\int_{0}^{2} x^{2} d x$ ?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

For $y = x^{2}$ on $[0, 2]$ , use two right rectangles.
3 marks
Width 1; heights 1 and 4 give 5.
$5$
For $y = x^{2}$ on $[0, 2]$ , use midpoint rectangles of width 1.
3 marks
Heights $f (0.5) = 0.25$ and $f (1.5) = 2.25$ sum to 2.5.
$5 / 2$
For $y = x^{2}$ on $[0, 2]$ , use four trapeziums.
3 marks
$h = 0.5$ ; heights 0,0.25,1,2.25,4.
$T = 0.25 [0 + 2 (0.25 + 1 + 2.25) + 4] = 2.75$ .
$11 / 4$
A rate is 4,6 and 10 L/min at 0,2 and 4 min. Estimate total volume by trapeziums.
3 marks
$T = 2 (4 + 6) / 2 + 2 (6 + 10) / 2 = 10 + 16 = 26$ .
26 L
The function $f (x) = 2 x + 1$ is sampled at 0,1 and 2. Is the trapezium estimate exact? Explain.
3 marks
A straight line matches each trapezium top exactly.
$T = (1 + 3) / 2 + (3 + 5) / 2 = 6$ .
Yes; its value is 6.
Why does the trapezium rule overestimate $\int_{0}^{2} x^{2} d x$ ?
3 marks
Each chord between neighbouring endpoints lies above the curve, so the trapezium includes extra area.
The graph is convex.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods integration and logarithmic antiderivatives; original learning and assessment material, with by-hand methods and explicitly described numerical models.

Official curriculum source ↗

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