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
Exam Tip
- If no diagram is provided, quickly sketch one so that you can see where the curve is above and below the x - axis and split up your integrals accordingly
- You should use your GDC to do this
Worked Example
The diagram below shows the graph of where.
The region is bounded by the curve, the -axis and the -axis.
The region is bounded by the curve, the x-axis and the line.
Find the total area of the shaded regions, and .
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
Exam Tip
- If no diagram is provided, quickly sketch one so that you can see where the curve is to the left and right of the y - axis and split up your integrals accordingly
- You should use your GDC to do this
Worked Example
Find the area enclosed by the curve with equation, the -axis and the horizontal lines with equations and .
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 -coordinate of any intersections between the curve and line
STEP 3
Use the graph to determine whether areas will need adding or subtracting
Deduce the limits and so the definite integral(s) to find the area(s) under hte curve and 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
Exam Tip
- Add information to any diagram provided
- Add axes intercepts, as well as intercepts between lines and curves
- Mark and shade the area you’re trying to find
- If no diagram provided, use your GDC to graph one and if you have time copy the sketch into your working
Worked Example
The region is bounded by the curve with equation and the line with equation.
lies entirely in the first quadrant.
Find the area of the region.