CIE AS Maths: Pure 1

Revision Notes

6.2.2 Area between a curve and a line

Test Yourself

Area between a curve and a line

What is the area between a curve and a straight line?

Notes fig1, AS & A Level Maths revision notes

  • This is the area enclosed by a curve and a line between their intersections
  • The intersections may need to be found using simultaneous equations

How do I find the area between a curve and a line?

  • The area enclosed will be the difference between …
    • the area under the curve and
    • the area under the line
  • These can be found separately

 Notes fig2, AS & A Level Maths revision notes

  • There may be easier ways to find the area under a line

 Notes fig3, AS & A Level Maths revision notes

  • Once the areas are found a question can be answered in full

Notes fig4, AS & A Level Maths revision notes

 

Notes fig5, AS & A Level Maths revision notes

 

Notes fig6, AS & A Level Maths revision notes

  • STEP 1: Find the intersections of the line and the curve
  • STEP 2: Find the area under a curve, RC, using definite integration
  • STEP 3: Find the area under a line, RL, either using definite integration or the area formulae for basic shapes
  • STEP 4: To find the area, R, between the curve and the line subtract the smaller area from the larger area
    • If curve on top this will be RC – RL
    • If line on top this will be RL - RC

Alternative method

  • An alternative method allows subtraction of the two functions first
  • But be extremely careful as to whether the curve or the line is on top

 

Notes fig7, AS & A Level Maths revision notes

  • STEP 1: Find the intersections of the line and the curve
  • STEP 2: Subtract the functions algebraically
    • Ensure this is done the correct way round!
  • STEP 3: Find the area, R, between the curve and the line using definite integration

Notes fig8, A Level & AS Level Pure Maths Revision NotesNotes fig9, AS & A Level Maths revision notes

Exam Tip

  • Add information to any diagram provided
  • Add axes intercepts, as well as intercepts between lines and curves
  • Mark and shade the area you’re trying to find
  • If no diagram provided, sketch one!

Worked example

Example Question fig1, A Level & AS Level Pure Maths Revision NotesExample fig2, A Level & AS Level Pure Maths Revision NotesExample fig3, AS & A Level Maths revision notesExample fig4, AS & A Level Maths revision notes

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Paul

Author: Paul

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.