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

Negative Integrals

• The area under a curve may appear fully or partially under the x-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

    • this occurs if the negative area(s) is/are greater than the positive area(s), their sum will be negative

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 all, or 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)

• This is given in the formula booklet

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 it