Definite Integrals & Areas (DP IB Applications & Interpretation (AI)): 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 

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

• 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 the -axis () and/or a root of

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

STEP 1

If a diagram is not given, use a GDC to draw the graph of
(or if already in that form)
If not identifiable from the question, use the graph to find the limits  and 

STEP 2

If needed, rearrange into the form

STEP 3

Write down the definite integral needed to find the required area

Use a GDC to evaluate it

A GDC is likely to require the function written with ‘’ as the variable (not ‘’)

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

Modulus may be called ‘Absolute value (Abs)’ on some GDCs

• 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

    • a GDC will apply  automatically as long as the modulus ( | ... | )symbols are included

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

    • using a GDC, one method is not particularly trickier than the other

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

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

STEP 1

If a diagram is not given, use a GDC to draw the graphs of the curve and line and identify the area to be found

STEP 2

Use a GDC to find the root(s) of the curve, the root of the line, and the x-coordinates of any intersections between the curve and the line.

STEP 3

Use the graph to determine whether areas will need adding or subtracting

Deduce the limits and thus the definite integral(s) to find the area(s) under the curve and the line
Use a GDC to calculate the area under the curve

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

Use a GDC to calculate the area under the line - this could be another definite integral or for a triangle or for a trapezium

STEP 4

Add or subtract areas accordingly to obtain a final answer

Definite Integrals

What is a definite integral?

• This is known as the Fundamental Theorem of Calculus

• a and b are called limits

  • a is the lower limit

  • b is the upper limit

• is the integrand

• is an antiderivative of

• The constant of integration (“+c”) is not needed in definite integration

  • “+c” would appear alongside both F(a) and F(b)

  • subtracting means the “+c”’s cancel

How do I find definite integrals analytically (manually)?

STEP 1

Give the integral a name to save having to rewrite the whole integral every time

If need be, rewrite the integral into an integrable form

STEP 2

Integrate without applying the limits; you will not need “+c”
Notation: use square brackets [ ] with limits placed at the end bracket

STEP 3

Substitute the limits into the function and evaluate