Area Between Curve & y-axis (DP IB Analysis & Approaches (AA)): Revision Note

Area Between Curve & y-axis

What is meant by the area between a curve and the y-axis?

• The area referred to is the region bounded by

  • the graph of 

  • the -axis

  • the horizontal line 

  • the horizontal line 

• The exact area can be found by evaluating a definite integral

• The graph of could be a straight line

  • using basic shape area formulae may be easier than integration

    • e.g. area of a trapezium: 

How do I find the area between a curve and the y-axis?

• Use the formula

  
  
  

  • This is given in the formula booklet

  • The function is normally given in the form 

  • so will need rearranging into the form 

  •  and may not be given directly as could involve the -axis () and/or a root of

    • use a GDC to plot the curve, sketch it and highlight the area to help

STEP 1
Identify the limits  and 
Sketch the graph of or use a GDC to do so, especially if and are not given directly in the question

STEP 2
Rearrange  into the form
This is similar to finding the inverse function

STEP 3
Evaluate the formula to evaluate the integral and find the area required
If using a GDC remember to include the modulus ( | … | ) symbols around  

• In trickier problems some (or all) of the area may be ‘negative’

  • this will be any area that is left of the -axis (negative -values)

  • || makes such areas ‘positive’

    • a GDC will apply ‘||’ automatically as long as the | … | are included

    • otherwise, to apply ‘||’, split the integral into positive and negative parts; write an integral and evaluate each part separately and add the modulus of each part together to give the total area