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

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

• Use the formula

• The function is normally given in the form

  • so will need rearranging into the form

•  and  may not be given directly and could involve the the -axis () and/or a root of (i.e. a point where the function intersects or touches the -axis)

  • You can use a GDC to plot the curve and find roots as necessary

• 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  

Can there be 'negative integrals' for areas between a curve and the y-axis?

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

  • This would be any area that is to the left of the -axis (negative  values)

  •  makes such areas 'positive' by reflecting them in the -axis

    • Your GDC will apply  automatically as long as you put the modulus symbol ( | ... | ) around

• Otherwise, to apply ‘’ manually, split the integral into positive and negative parts

  • Write an integral and evaluate each part separately

  • Change any negative values found to positive

  • Then add all the positive values together to give the total area