Applications of Integration (DP IB Maths: AI HL)

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Paul

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Paul

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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 c, 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 straight F left parenthesis x right parenthesis plus c equals integral f left parenthesis x right parenthesis space straight d x, the constant of integration, c - and so the particular antiderivative - can be found if a point the graph of y equals straight F left parenthesis x right parenthesis plus c passes through is known 

 STEP 1
 If need be, rewritespace f left parenthesis x right parenthesis into an integrable form
 Each term needs to be a power of x (or a constant)
 
 STEP 2
 Integrate each term ofspace f apostrophe left parenthesis x right parenthesis, remembering the constant of integration, “plus c
 (Increase power by 1 and divide by new power)
 
 STEP 3
Substitute the x and y coordinates of a given point in to straight F left parenthesis x right parenthesis plus c to form an equation in c
Solve the equation to find c

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 y equals f left parenthesis x right parenthesis passes through the point left parenthesis 3 comma negative 4 right parenthesis.  The gradient function ofspace f left parenthesis x right parenthesis is given byspace f apostrophe left parenthesis x right parenthesis equals 3 x squared minus 4 x minus 4.

Findspace f left parenthesis x right parenthesis.

5-2-3-ib-sl-ai-aa-we1-soltn-

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Area Under a Curve Basics

What is meant by the area under a curve?

Notes fig1

  • The phrase “area under a curve” refers to the area bounded by
    • the graph of y equals f left parenthesis x right parenthesis
    • the x-axis
    • the vertical line x equals a
    • the vertical line x equals b
  • The exact area under a curve is found by evaluating a definite integral
  • The graph of y equals f left parenthesis x right parenthesis 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 “A equals 1 half h left parenthesis a plus b right parenthesis

What is a definite integral?

integral subscript a superscript b f left parenthesis x right parenthesis space straight d x equals straight F left parenthesis b right parenthesis minus straight F left parenthesis a right parenthesis

  • 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
  • space f left parenthesis x right parenthesis is the integrand
  • straight F left parenthesis x right parenthesis is an antiderivative ofspace f left parenthesis x right parenthesis
  • The constant of integration (“plus c”) is not needed in definite integration
    • "plus c” would appear alongside both F(a) and F(b)
    • subtracting means the “plus c”’s cancel

How do I form a definite integral to find the area under a curve?

  • The graph of y equals f left parenthesis x right parenthesis and the x-axis should be obvious boundaries for the area so the key here is in finding a and b - 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 y-axis being one of the (vertical) boundaries – in this case the limit (x) will be 0
One, or both, of the limits, could be a root of the equationspace f left parenthesis x right parenthesis equals 0
i.e.  where the graph ofspace y equals f left parenthesis x right parenthesis crosses the x-axis
In this case solve the equationspace f left parenthesis x right parenthesis equals 0 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
space A equals integral subscript a superscript b f left parenthesis x right parenthesis space straight d x

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

integral subscript square superscript square square

  • 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) (space f left parenthesis x right parenthesis)
    • The lower limit (a from x equals a)
    • The upper limit (b from x equals b)
  • 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 y equals f left parenthesis x right parenthesis
    • You may also wish to plot the vertical lines x equals a and x equals b
      • make sure your GDC is expecting an "x equalsstyle 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. integral straight d x)
      • 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

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
y equals x to the power of 4 minus 2 x squared plus 5,
x equals 1 and
x equals 2 on the same diagram

5-2-3-ib-sl-ai-aa-we2-soltn-ab)
The area enclosed by the three graphs from part (a) and the x-axis is to be found.
Write down an integral that would find this area.
5-2-3-ib-sl-ai-aa-we2-soltn-b
c)
Using your GDC, or otherwise, find the exact area described in part (b).
Give your answer in the form a over b where a and b are integers.
5-2-3-ib-sl-ai-aa-we2-soltn-c

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.