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Applications of Integration (DP IB Maths: AI HL)
Revision Note
Finding the Constant of Integration
What is the constant of integration?
- When finding an anti-derivative there is a constant term to consider
- this constant term, usually called , is the constant of integration
- 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?
- For , the constant of integration, - and so the particular antiderivative - can be found if a point the graph of passes through is known
STEP 1
If need be, rewrite into an integrable form
Each term needs to be a power of (or a constant)
STEP 2
Integrate each term of, remembering the constant of integration, “”
(Increase power by 1 and divide by new power)
STEP 3
Substitute the and coordinates of a given point in to to form an equation in
Solve the equation to find
Examiner 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
The graph of passes through the point . The gradient function of is given by.
Find.
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Area Under a Curve Basics
What is meant by the area under a curve?
- The phrase “area under a curve” refers to the area bounded by
- the graph of
- the -axis
- the vertical line
- the vertical line
- The exact area under a curve is found by evaluating a definite integral
- The graph of could be a straight line
- the use of integration described below would still apply
- but the shape created would be a trapezoid
- so it is easier to use “”
- the use of integration described below would still apply
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 (“”) is not needed in definite integration
- "” would appear alongside both F(a) and F(b)
- subtracting means the “”’s cancel
How do I form a definite integral to find the area under a curve?
- The graph of and the -axis should be obvious boundaries for the area so the key here is in finding and - the lower and upper limits of the integral
STEP 1
Use the given sketch to help locate the limits
You may prefer to plot the graph on your GDC and find the limits from there
STEP 2
Look carefully where the ‘left’ and ‘right’ boundaries of the area lie
If the boundaries are vertical lines, the limits will come directly from their equations
Look out for the -axis being one of the (vertical) boundaries – in this case the limit () will be 0
One, or both, of the limits, could be a root of the equation
i.e. where the graph of crosses the -axis
In this case solve the equation to find the limit(s)
A GDC will solve this equation, either from the graphing screen or the equation solver
STEP 3
The definite integral for finding the area can now be set up in the form
Examiner 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!
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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
- The function to be integrated (integrand) ()
- The lower limit ( from )
- The upper limit ( from )
- 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 ...
- Plot the graph of
- You may also wish to plot the vertical lines and
- make sure your GDC is expecting an "" style equation
- Once you have plotted the graph you need to look for an option regarding “area” or a physical button
- it may appear as the integral symbol (e.g. )
- 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
- You may also wish to plot the vertical lines and
Examiner 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
a)
Using your GDC to help, or otherwise, sketch the graphs of
,
and
on the same diagram
b)
The area enclosed by the three graphs from part (a) and the -axis is to be found.
Write down an integral that would find this area.
c)
Using your GDC, or otherwise, find the exact area described in part (b).
Give your answer in the form where and are integers.
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