GraphsStudy Notes

Worked Examples

Part of Area Under Curves — GCSE Mathematics

This study notes covers Worked Examples 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 9 of 11 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 9 of 11

Practice

9 questions

Recall

10 flashcards

✏️ Worked Examples

Example 1: Applying the Trapezium Rule

Question: Use the trapezium rule with 4 strips to estimate the area under a curve where the y-values at x = 0, 2, 4, 6, 8 are: y = 1, 4, 9, 7, 3. The strip width h = 2.

Show Solution

Step 1: Identify first, last, and middle y-values

First: y₀ = 1 Last: y₄ = 3 Middle: y₁ = 4, y₂ = 9, y₃ = 7

Step 2: Apply the trapezium rule formula

Area ≈ h/2 × (y₀ + 2y₁ + 2y₂ + 2y₃ + y₄)

Area ≈ 2/2 × (1 + 2(4) + 2(9) + 2(7) + 3)

Area ≈ 1 × (1 + 8 + 18 + 14 + 3)

Area ≈ 1 × 44 = 44 square units

Answer: Estimated area ≈ 44 square units

Example 2: Area Under a Speed-Time Graph

Question: A speed-time graph has 3 equal strips of width h = 5 seconds. The speed values at t = 0, 5, 10, 15 seconds are: 10, 14, 12, 8 m/s. Estimate the total distance travelled.

Show Solution

Step 1: Identify values — h = 5, y₀ = 10, y₁ = 14, y₂ = 12, y₃ = 8

Step 2: Apply the trapezium rule

Distance ≈ h/2 × (y₀ + 2y₁ + 2y₂ + y₃)

Distance ≈ 5/2 × (10 + 2(14) + 2(12) + 8)

Distance ≈ 2.5 × (10 + 28 + 24 + 8)

Distance ≈ 2.5 × 70 = 175 m

Interpretation: The object travels approximately 175 metres in 15 seconds.

Answer: Estimated distance ≈ 175 m

Practice Questions for Area Under Curves

On a velocity-time graph, what does the area under the curve represent?

A. The acceleration
B. The average velocity
C. The distance (or displacement) travelled
D. The speed at a given moment

1 markfoundation

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.

2 markshigher

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).