y = f(x) is above y = g(x) between x = a and x = b. Which gives the area between them?

Areas between curves, and areas below the x-axis

Integration finds the area between a curve and a line, or between two curves. Parts of a curve below the x-axis need special care.

International GCSE · 4PM1 · 9C
About 18 minutes

Practise this lesson free (6 questions)

Between two graphs

Area = ∫_a^b (top − bottom) dx, where a and b are where the graphs meet.
y = 2x and y = x² meet where x² = 2x, at x = 0 and x = 2. The line is on top.
Area = ∫₀² (2x − x²) dx = x² − x³<span>3₀² = 4 − 83 = 43

Below the x-axis

Where the curve is below the x-axis, the integral comes out negative. The area is its size.
y = x² − 4 from 0 to 3 crosses the axis at x = 2.
∫₀² = −163 and ∫₂³ = 73.
Total area = 163 + 73 = 233, but ∫₀³ on its own gives −3, which is wrong.

Method

1. Sketch both graphs.
2. Find where they meet (solve simultaneously) and where the curve crosses the axis.
3. Integrate top minus bottom between each pair of limits.
4. Add the sizes of the separate areas.

Worked example

Find the area enclosed by y = 4 − x² and y = x + 2.

Meet where 4 − x² = x + 2: x² + x − 2 = 0, x = -2 or 1
The curve is on top: ∫ (4 − x² − x − 2) dx = ∫ (2 − x − x²) dx
2x − x²<span>2 − x³3 from -2 to 1 = 92

Answer: 92

Key idea

Area between graphs: integrate top minus bottom between the meeting points. Split at the x-axis and add the sizes of the parts.

Quick check

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Extra practice

2 more questions

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Find the area enclosed by y = 4x and y = x². Give your answer as a fraction in its simplest form.
Show the answer
Answer: 32/3
They meet at x = 0 and x = 4. ∫₀⁴ (4x − x²) dx = 323.
Find the area enclosed by y = 6 − x² and y = x. Give your answer as a fraction in its simplest form.
Show the answer
Answer: 125/6
They meet at x = −3 and x = 2. ∫ (−x² − x + 6) dx between these = 1256.

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Practise this lesson free (6 questions)All International GCSE Further Pure Mathematics lessons

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Sources and references

Pearson Edexcel International GCSE Further Pure Mathematics (4PM1) specification, topic 9C

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.