Estimating gradients and areas under graphs

For real-context curves, WJEC Higher requires both gradient (rate) and area (cumulative quantity) interpretations.

Estimating area under a curve

Two common methods on WJEC papers:

1. Trapezium rule (informal)

Divide the interval into equal-width strips and approximate each strip by a trapezium.

For n strips of width h, with heights y₀, y₁, ..., yₙ: Area ≈ h/2 × (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ).

Equivalently: h × [(first + last)/2 + middle heights].

2. Counting squares

Each grid square has known area. Count whole squares; estimate part-squares as halves or quarters. Sum.

This is acceptable on Foundation/Intermediate.

Real-world meaning of "area under"

• Speed–time curve: area = distance travelled.
• Acceleration–time curve: area = change in velocity.
• Flow rate (litres/s) vs time (s): area = total volume in litres.
• Power (kW) vs time (h): area = energy in kWh.

Gradient at a point — recap

Tangent at the point; gradient = rise/run on the tangent.

Combined gradient + area question

A typical WJEC Higher paper asks: (a) Use a tangent at t = 4 to estimate the acceleration. (3 marks) (b) Estimate the total distance travelled over 0 ≤ t ≤ 8 using the trapezium rule with 4 strips. (4 marks)

Accuracy

• Tangents: ±0.5 typical tolerance on the gradient value.
• Trapezium rule: under-estimates area for concave-up curves, over-estimates for concave-down curves. WJEC accepts ± 5% from the published answer.

Worked trapezium example

Speed–time curve: at t = 0, 2, 4, 6, 8 the speeds are 0, 12, 18, 22, 24 m/s. Strip width h = 2.

Area ≈ 2/2 × (0 + 2(12) + 2(18) + 2(22) + 24) = 1 × (0 + 24 + 36 + 44 + 24) = 128 m.

So total distance ≈ 128 m.

WJEC exam tip

When using the trapezium rule, list the y-values in a table BEFORE substituting. This gives you a clean record M1 and lets the examiner trace any arithmetic slip back to a correct method.