Finding Areas (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

• When using a GDC use the modulus (absolute value) function so that all definite integrals have a positive value

• This is given in the formula booklet

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 the curve is partially under the x-axis?

• For questions that allow the use of a GDC you can still use

• To find the area analytically (manually) use the following method

STEP 1

Split the area into parts - the area(s) that are above the x-axis and the area(s) that are below the x-axis

 STEP 2

Write the expression for the definite integral for each part (give each part a name, I₁, I₂, etc)

This may involve finding the lower and upper limits of each part from a graph sketch or a GDC, finding the roots of the function (i.e. where) and rewriting in an integrable form

 STEP 3

Find the value of each definite integral separately

 STEP 4

Find the area by summing the modulus (absolute values) of each integral

(Mathematically this would be written)         

Area Between a Curve and a Line

• Areas whose boundaries include a curve and a (non-vertical) straight line can be found using integration

  • For an area under a curve a definite integral will be needed

  • For an area under a line the shape formed will be a trapezium or triangle

    • basic area formulae can be used rather than a definite integral

    • (although a definite integral would still work)

• The area required could be the sum or difference of areas under the curve and line

How do I find the area between a curve and a line?

STEP 1

If not given, sketch the graphs of the curve and line on the same diagram

Use a GDC to help with this step

STEP 2

Find the intersections of the curve and the line

If no diagram is given this will help identify the area(s) to be found

 STEP 3

Determine whether the area required is the sum or difference of the area under the curve and the area under the line
Calculate the area under a curve using a integral of the form

Calculate the area under a line using either for a triangle or for a trapezium (y-coordinates will be needed)

 STEP 4

Evaluate the definite integrals and find their sum or difference as necessary to obtain the area required

Area Between 2 Curves

• Areas whose boundaries include two curves can be found by integration

  • The area between two curves will be the difference of the areas under the two curves

    • both areas will require a definite integral

  • Finding points of intersection may involve a more awkward equation than solving for a curve and a line

How do I find the area between two curves?

STEP 1

If not given, sketch the graphs of both curves on the same diagram

Use a GDC to help with this step

 STEP 2

Find the intersections of the two curves

If no diagram is given this will help identify the area(s) to be found

STEP 3

For each area (there may only be one) determine which curve is the ‘upper’ boundary

For each area, write a definite integral of the form

where is the function for the ‘upper’ boundary and is the function for the ‘lower’ boundary

Be careful when there is more than one region – the ‘upper’ and ‘lower’ boundaries will swap

 STEP 4

Evaluate the definite integrals and sum them up to find the total area

(Step 3 means no definite integral will have a negative value)