Finding Areas Between Lines & Curves (Cambridge O Level Additional Maths)

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

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