Gradients and areas under graphs
OCR J560 Higher (J560/04–06) tests numerical estimation of gradients of curves and areas under curves, with interpretation in context. This is the GCSE bridge to integration and differentiation.
Estimating a gradient at a point
(See R15 for the tangent method.) Draw a tangent at the point of interest, pick two clear points on the tangent, and compute (Δy)/(Δx). Quote the answer with units appropriate to the axes.
Estimating an area under a curve
For a straight-line graph, area = a triangle, rectangle, or trapezium.
For a curve, use the trapezium rule: split the area into vertical strips of equal width h. Each strip is approximated by a trapezium. Sum the areas.
Trapezium rule (n strips of width h): Area ≈ (h/2) × [y₀ + 2(y₁ + y₂ + ... + y_{n−1}) + y_n]
Where y₀, y₁, ..., y_n are the y-values at the strip boundaries.
✦Worked example— Worked example — trapezium rule
Estimate the area under y = x² between x = 0 and x = 4 using 4 strips of width 1.
Boundaries: x = 0, 1, 2, 3, 4 with y = 0, 1, 4, 9, 16.
Area ≈ (1/2) × [0 + 2(1 + 4 + 9) + 16] = (1/2) × [0 + 28 + 16] = (1/2) × 44 = 22.
The exact area is 64/3 ≈ 21.33, so the trapezium rule overestimates here (curve is below the chords).
Interpretation
On a speed–time graph, area = distance. On a flow rate–time graph (rate of water in litres/s vs time), area = total volume. On a current–time graph (amperes vs time), area = total charge in coulombs.
Over- vs under-estimate
The trapezium rule:
- Overestimates when the curve is concave up (cup-shaped, like y = x²).
- Underestimates when the curve is concave down (cap-shaped).
OCR Higher questions sometimes ask you to state which.
OCR mark scheme conventions
- M1 for splitting the region into trapezia of equal width.
- M1 for substituting y-values correctly.
- A1 for the numerical area.
- B1 for the contextual interpretation with units (e.g. "the total distance is 22 m").
⚠Common mistakes
- Using unequal strip widths without adjusting the formula.
- Forgetting the factor 1/2 in the trapezium rule.
- Doubling the endpoint y-values (only the inner ones get doubled).
- Reporting an area without context or units.
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