IB DP Maths: AI HL复习笔记5.3.3 Applications of Integration
Finding the Constant of Integration
What is the constant of integration?
- When finding an anti-derivative there is a constant term to consider
In terms of graphing an anti-derivative, there are endless possibilities
- collectively these may be referred to as the family of antiderivatives or family of curves
- the constant of integration is determined by the exact location of the curve
- if a point on the curve is known, the constant of integration can be found
How do I find the constant of integration?
Exam Tip
- If a constant of integration can be found then the question will need to give you some extra information
- If this is given then make sure you use it to find the value of c
Worked Example
Area Under a Curve Basics
What is meant by the area under a curve?
What is a definite integral?
How do I form a definite integral to find the area under a curve?
Exam Tip
- Look out for questions that ask you to find an indefinite integral in one part (so “+c” needed), then in a later part use the same integral as a definite integral (where “+c” is not needed)
- Add information to any diagram provided in the question, as well as axes intercepts and values of limits
- Mark and shade the area you’re trying to find, and if no diagram is provided, sketch one!
Definite Integrals using GDC
Does my calculator/GDC do definite integrals?
- Modern graphic calculators (and some ‘advanced’ scientific calculators) have the functionality to evaluate definite integrals
- i.e. they can calculate the area under a curve (see above)
- If a calculator has a button for evalutaing definite integrals it will look something like
- This may be a physical button or accessed via an on-screen menu
- Some GDCs may have the ability to find the area under a curve from the graphing screen
- Be careful with any calculator/GDC, they may not produce an exact answer
How do I use my GDC to find definite integrals?
Without graphing first …
- Once you know the definite integral function your calculator will need three things in order to evaluate it
Have a play with the order in which your calculator expects these to be entered – some do not always work left to right as it appears on screen!
With graphing first ...
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-
- your GDC may allow you to select the lower and upper limits by moving a cursor along the curve - however this may not be very accurate
- your GDC may allow you to type the exact limits required from the keypad
- the lower limit would be typed in first
- read any information that appears on screen carefully to make sure
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Exam Tip
- When revising for your exams always use your GDC to check any definite integrals you have carried out by hand
- This will ensure you are confident using the calculator you plan to take into the exam and should also get you into the habit of using you GDC to check your work, something you should do if possible
Worked Example
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