This introduction covers Measuring What You Cannot Calculate Directly 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 1 of 11 in this topic. Use this introduction to connect the idea to the wider topic before moving on to questions and flashcards.
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.
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).