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Measuring What You Cannot Calculate Directly
Measuring What You Cannot Calculate Directly
A car's speedometer tells you how fast you are going right now. But how far did you travel during a journey where the speed was constantly changing — speeding up, slowing down, stopping? You cannot simply multiply speed by time because the speed was not constant. Instead, you find the area under the speed-time graph. Even if the graph is curved, the area still gives the total distance. Estimating this area using the trapezium rule is one of the most practically useful techniques in GCSE mathematics.
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.
Mathematics glossary
- 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).
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💡 Exam Tips for Area Under Curves
- State the y-values clearly in a table before applying the formula — this avoids errors
- First and last once, all others twice — the classic memory device for the trapezium rule
- Check the strip width h — it must be the same for all strips when using the standard formula
- Give units: area on a speed-time graph has units of distance (m, km); always state the units of your answer
- Over/underestimate: look at whether the curve bends up (under) or down (over) relative to the trapezium tops
- More strips = better estimate — halving h approximately halves the error
Now try it yourself
Quiz · Question 1 of 9
On a velocity-time graph, what does the area under the curve represent?
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