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A LevelMaths: PureAQARevision Notes8. Integration8.1 Integration8.1.4 Area Under a Curve

Area Under a Curve (AQA A Level Maths: Pure)

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Paul

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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 Tip

  • 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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    Previous:8.1.3 Definite IntegrationNext:8.1.5 Area between a curve and a line
    Paul

    Author: Paul

    Expertise: Maths

    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 – one of the many reasons he is excited to be a member of the SME team.