Finding Areas Between Lines & Curves (Cambridge (CIE) O Level Additional Maths): Revision Note

Area Between a Curve & a Line

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

• The area enclosed will be the difference between …

  • the area under the line and

  • the area under the curve

• These can be found separately

• STEP 1

  •  Find the intersections of the line and the curve (if not given)

• STEP 2

  • Find the area under a curve, R_C, using definite integration

• STEP 3 

  • Find the area under a line, R_L, 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 R_C – R_L

  • If line on top this will be R_L - R_C

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

Can I subtract before integrating?

• Yes - see the next note on "Area between two curves"

• Essentially you treat the line as a second curve 

  • It can be easier to make mistakes using this method

Area Between Two Curves

What is the area between two curves?

• In general find the definite integral of “upper curve” – “lower curve”

• However this does depend on …

  • … the area being found

  • … if the curves intersect (and cross over)

• The area may have to be split into separate integrals

• The points at which curves intersect may need to be calculated

How do I find the area between two curves?

• STEP 1

  • Find the intersections of the curves if needed

• STEP 2

  •  Form the integral …

    • … using the intersections as limits

    • … “upper curve” – “lower curve” …

    • … and find the value of the integral

• STEP 3

  • Repeat STEP 2 if more than one area needed

• STEP 4

  • Add areas together