This deep dive covers Overestimate or Underestimate? 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 6 of 11 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Overestimate or Underestimate?
The trapezium rule uses straight-line tops instead of the actual curve, so it gives an approximation, not the exact answer.
- Underestimate: when the curve is concave UPWARD (curves up, like a bowl) — the straight tops of the trapezoids lie BELOW the curve
- Overestimate: when the curve is concave DOWNWARD (curves down, like a hill) — the straight tops lie ABOVE the curve
- To improve accuracy: use more strips (narrower h). The estimate gets closer to the exact area as h → 0
Exam tip: If asked "why is this an underestimate?", say: "The curve is concave upward, so the straight-line tops of the trapeziums lie below the actual curve, missing some area."
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.