Negative Integrals (DP IB Analysis & Approaches (AA)): Revision Note

Negative Integrals

What do we mean by 'negative integrals'?

• The area between a curve and the -axis may appear fully or partially under the -axis

  • This occurs when the function takes negative values within the boundaries of the area

• The definite integrals used to find such areas will be

  • negative if the area is fully under the-axis

  • possibly negative if the area is partially under the-axis

    • In neither case will the integral give you the (positive) answer you are looking for for the area

How do I find the area under a curve when the curve is fully under the x-axis?

• STEP 1

  Write the expression for the definite integral to find the area as usual

  • This may involve finding the lower and upper limits from a graph sketch or GDC and f(x) may need to be rewritten in an integrable form

• STEP 2

  The answer to the definite integral will be negative

  Area must always be positive so take the modulus (absolute value) of it

  • e.g.  If then the area would be 36 (square units)

How do I find the area under a curve when some of the curve is below the x-axis?

• Use the modulus function

  • The modulus is also called the absolute value (Abs)

  • Essentially the modulus function makes all function values positive

  • Graphically, this means any negative areas are reflected in the -axis

• A GDC will recognise the modulus function

  • look for a key or on-screen icon that says 'Abs' (absolute value)

• STEP 1

  If a diagram is not given, use a GDC to draw the graph of

  If not identifiable from the question, use the graph to find the limits  and 
   

• STEP 2

  Write down the definite integral needed to find the required area

  Remember to include the modulus ( | ... | ) symbols around the function

  Use the GDC to evaluate the integral

• You can also apply ‘’ manually, by splitting 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