Area Under a Curve (Cambridge (CIE) A Level Maths) : Revision Note

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Area Under a Curve

What does area under a curve mean?

  • The phrase “area under a curve” refers to the area bounded by …

    • ... the x-axis

    • … the graph of y = f(x)

    • … the vertical line x = a

    • … the vertical line x = b

Notes fig1, AS & A Level Maths revision notes

How do I find the area under a curve?

Notes fig2, AS & A Level Maths revision notes
  • The value from definite integration is equal to the “area under a curve”

    Notes fig3, AS & A Level Maths revision notes

What if I am not told the limits?

Notes fig4, AS & A Level Maths revision notes
  •  If limits are not provided they will be the x-axis intercepts

    • Set y = 0 and solve the equation to find the x-axis intercepts

Notes fig5, AS & A Level Maths revision notes

Negative areas

Notes fig6, A Level & AS Level Pure Maths Revision Notes
  •  If the area lies underneath the x-axis the value of the integral will be negative

  • An area cannot be negative, so take the modulus of the integral

Notes fig7, AS & A Level Maths revision notes

What if the area is made up of more than one section?

Notes fig8, A Level & AS Level Pure Maths Revision Notes
  • Be careful when one section is above and one section is below the x-axis

    • You will need a separate integral for each section, BUT...

    • ...One section's integral will be negative

    • ...One section's integral will be positive

    • So you’ll need to take the modulus before adding to find the total area

    Notes fig9, A Level & AS Level Pure Maths Revision Notes

Examiner Tips and Tricks

  • Add information to any diagram provided in the question, as well as axes intercepts and values of limits.

  • Mark and shade the area you’re trying to find, and if no diagram is provided, sketch one!

Worked Example

Example fig1, AS & A Level Maths revision notes
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Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

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