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11.8PureStretch

Finding Areas

A definite integral gives the signed area between a curve and the x-axis. To find a true area you must handle regions below the axis and areas between two curves carefully.

26 min Integration
Edexcel A level a Maths: 11.8 Finding Areas (Part 1)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Find the area under a curve by definite integration
  • Handle areas below the x-axis
  • Find the area between two curves
  • Combine regions correctly
1

Area under a curve

The area between y=f(x)y = f(x), the xx-axis and x=a,x=bx = a, x = b is ∫abf(x) dx\displaystyle\int_a^b f(x)\,dx — provided the curve is above the axis throughout.

Area=∫abf(x) dx\text{Area} = \int_a^b f(x)\,dx
Area under a curve (above the axis).
2

Below the axis and between curves

Where the curve is below the axis the integral is negative, so split the region and take the modulus of each part. The area between two curves is ∫ab(upper−lower) dx\displaystyle\int_a^b (\text{upper} - \text{lower})\,dx.

Example 1Area between y=xy = x and y=x2y = x^2 from 0 to 1.
1Upper is y=xy = x. ∫01(x−x2) dx\int_0^1 (x - x^2)\,dx.
2=[x22−x33]01=12−13=16= [\tfrac{x^2}{2} - \tfrac{x^3}{3}]_0^1 = \tfrac12 - \tfrac13 = \tfrac16.
Answer
16\tfrac16

Tip — A negative integral means the region is below the axis — take its modulus for the area.

Formula recap

Area=∫abf(x) dx\text{Area} = \int_a^b f(x)\,dx
Above the axis.
∫ab(upper−lower) dx\int_a^b (\text{upper} - \text{lower})\,dx
Between two curves.

Common mistakes to avoid

Treating a negative integral as a negative area.
Area is positive; take the modulus of the below-axis part.
Subtracting lower from upper the wrong way.
Always upper minus lower for area between curves.

Key takeaways

  • Area under a curve = ∫ₐᵇ f(x) dx (above axis).
  • Below the axis: take the modulus of that part.
  • Between curves: ∫(upper − lower) dx.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The area under y = f(x) from a to b (above axis) is:

    • ∫abf(x) dx\int_a^b f(x)\,dx
    • f(b)−f(a)f(b) - f(a)
    • ∫abf′(x) dx\int_a^b f'(x)\,dx
    • f(a)+f(b)f(a) + f(b)
    Show answer and explanation

    Answer: ∫abf(x) dx\int_a^b f(x)\,dx

    Definite integral.

  2. A negative definite integral indicates the region is:

    • below the x-axis
    • above the axis
    • a maximum
    • undefined
    Show answer and explanation

    Answer: below the x-axis

    Signed area.

  3. The area between two curves is:

    • ∫(upper−lower) dx\int (\text{upper}-\text{lower})\,dx
    • ∫(lower−upper) dx\int (\text{lower}-\text{upper})\,dx
    • ∫upper dx\int \text{upper}\,dx
    • their sum
    Show answer and explanation

    Answer: ∫(upper−lower) dx\int (\text{upper}-\text{lower})\,dx

    Top minus bottom.

  4. Area between y=xy=x and y=x2y=x^2 from 0 to 1 is:

    • 16\tfrac16
    • 12\tfrac12
    • 13\tfrac13
    • 11
    Show answer and explanation

    Answer: 16\tfrac16

    1/2 − 1/3.

  5. For a region partly below the axis you should:

    • split and take moduli
    • integrate straight through
    • ignore the negative part
    • differentiate
    Show answer and explanation

    Answer: split and take moduli

    Avoid cancellation.

Next step

Check your understanding

Work through questions on Finding Areas. Use the feedback to decide what needs another look.

1 past-paper question · 8 marks

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