This topic summary covers Knowledge Organiser: Area Under Curves within Area Under Curves for GCSE Mathematics. Revise Area Under Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 11 of 11 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Area Under Curves
Trapezium Rule Formula
- Area ≈ h/2 × (y₀ + 2y₁ + ... + 2yₙ₋₁ + yₙ)
- h = equal strip width
- First and last y: appear once
- All middle y-values: multiply by 2
Accuracy and Context
- More strips → more accurate estimate
- Concave up curve → underestimate
- Concave down curve → overestimate
- Speed-time: area = distance travelled
- More strips = better approximation of the true area
Key Vocabulary
- Trapezium rule: estimate using strips with straight tops
- Strip width (h): equal horizontal width of each strip
- Overestimate: trapezoids exceed actual area
- Underestimate: trapezoids miss some area below curve
- Approximation: the trapezium rule gives an estimate, not an exact answer
Common Errors
- Multiplying first/last y-values by 2 (they appear once)
- Thinking trapezium rule gives exact answer
- Forgetting units on area answer
- Confusing distance (|area|) with displacement (signed area)
Key Formulas
- Trapezium rule: A ≈ ½h(y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ)
- h = (b − a) ÷ n (strip width)
- More strips → more accurate estimate
- Area under velocity-time graph = distance travelled
Practice questions for Area Under Curves
On a velocity-time graph, what does the area under the curve represent?
Explain whether the trapezium rule gives an overestimate or underestimate for the area under the curve y = x² between x = 0 and x = 3. Justify your answer.
Quick recall flashcards
What is the trapezium rule used for?
Estimating the area under a curve by dividing it into trapezium-shaped strips.
Each strip is a trapezium (not a rectangle or triangle).
The more strips used, the more accurate the estimate.
It is an approximation — the actual area is slightly different because the tops of the strips are straight lines, not curves.
What does the area under a speed-time graph represent?
Distance travelled.
Area under a speed-time graph = distance travelled
This works for any speed-time or velocity-time graph.
Even if the graph is curved, the area still represents distance.
Units: if speed is m/s and time is s, then area has units of metres (m).