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IB DP Maths: AA HL复习笔记5.4.4 Further Applications of Integration

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月14日下午6:04

Negative Integrals

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

- 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

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

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

Worked Example

5-4-4-ib-sl-aa-only-we1-soltn-a

5-4-4-ib-sl-aa-only-we1-soltn-c

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

STEP 4

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

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 is provided, sketch one!

Worked Example

5-4-4-ib-sl-aa-only-we2-soltn-a

5-4-4-ib-sl-aa-only-we2-soltn-b

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

8.2.10-Notes-curv_hard

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

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)

Exam Tip

If no diagram is provided sketch one, even if the curves are not accurate
Add information to any given diagram as you work through a question
Maximise use of your GDC to save time and maintain accuracy:
- Use it to sketch the graphs and help you visualise the problem
- Use it to find definite integrals