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Finding Areas with Integration (CIE IGCSE Additional Maths)

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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

Example of area under a curve

How do I find the area under a curve?

  • The value from definite integration is equal to the “area under a curve”Definite integration gives the area under a curveExample of finding the area under a curve

What if I am not told the limits?

  •  If limits are not provided they will likely be the x-axis intercepts
    • Set y = 0 and solve the equation to find the x-axis intercepts

Example of finding the limits of integration

What happens if the graph is below the x-axis?

  •  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 integralExample of the area being below the x-axis

Another example of the area being below the x-axis

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

  • 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

Splitting the area into two areas

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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    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.