Definite Integrals (DP IB Maths: AA HL)

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Paul

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Definite Integrals

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

How do I find definite integrals analytically (manually)?

STEP 1
Give the integral a name to save having to rewrite the whole integral every time
If need be, rewrite the integral into an integrable form

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

STEP 2
Integrate without applying the limits; you will not need “+c
Notation: use square brackets [ ] with limits placed at the end bracket
 
STEP 3
Substitute the limits into the function and evaluate

Exam Tip

  • If a question does not state that you can use your GDC then you must show all of your working clearly, however it is always good practice to check you answer by using your GDC if you have it in the exam

Worked example

a)
Show that

integral subscript 2 superscript 4 3 x left parenthesis x squared minus 2 right parenthesis space straight d x equals 144

 5-4-3-ib-sl-aa-only-we1-soltn-a

b)
Use your GDC to evaluate

space integral subscript 0 superscript 1 3 e to the power of x squared sin space x end exponent space straight d x

giving your answer to three significant figures.

5-4-3-ib-sl-aa-only-we1-soltn-b

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Properties of Definite Integrals

Fundamental Theorem of Calculus

  • Formally,
    • space straight f left parenthesis x right parenthesis is continuous in the intervalspace a less or equal than x less or equal than b
    • space straight F left parenthesis x right parenthesisis an antiderivative ofspace straight f left parenthesis x right parenthesis

What are the properties of definite integrals?

  • Some of these have been encountered already and some may seem obvious …
    • taking constant factors outside the integral
      • integral subscript a superscript b k straight f left parenthesis x right parenthesis space straight d x equals k integral subscript a superscript b straight f left parenthesis x right parenthesis space straight d x wherespace k is a constant
      • useful when fractional and/or negative values involved
    • integrating term by term
      • space integral subscript a superscript b left square bracket straight f left parenthesis x right parenthesis plus straight g left parenthesis x right parenthesis right square bracket space straight d x equals integral subscript a superscript b straight f left parenthesis x right parenthesis space straight d x plus integral subscript a superscript b straight g left parenthesis x right parenthesis space straight d x 
      • the above works for subtraction of terms/functions too
    • equal upper and lower limits
      • integral subscript a superscript a straight f left parenthesis x right parenthesis space d x equals 0 
      • on evaluating, this would be a value, subtract itself !
    • swapping limits gives the same, but negative, result
      • integral subscript a superscript b straight f left parenthesis x right parenthesis space straight d x equals negative integral subscript b superscript a straight f left parenthesis x right parenthesis space straight d x 
      • compare 8 subtract 5 say, with 5 subtract 8 …
    • splitting the interval
      • space integral subscript a superscript b straight f left parenthesis x right parenthesis space straight d x equals integral subscript a superscript c straight f left parenthesis x right parenthesis space straight d x plus integral subscript c superscript b straight f left parenthesis x right parenthesis space straight d x wherespace a less or equal than c less or equal than b
      • this is particularly useful for areas under multiple curves or areas under thespace x-axis
    • horizontal translations
      • space integral subscript a superscript b straight f left parenthesis x right parenthesis space straight d x equals integral subscript a minus k end subscript superscript b minus k end superscript straight f left parenthesis x plus k right parenthesis space straight d x wherespace k is a constant
      • the graph ofspace y equals straight f left parenthesis x plus-or-minus k right parenthesis is a horizontal translation of the graph ofspace y equals straight f left parenthesis x right parenthesis
        (straight f left parenthesis x plus k right parenthesis translates left, straight f left parenthesis x minus k right parenthesis translates right)

Exam Tip

  • Learning the properties of definite integrals can help to save time in the exam

Worked example

space straight f left parenthesis x right parenthesis is a continuous function in the intervalspace 5 less or equal than x less or equal than 15 .

It is known thatspace integral subscript 5 superscript 10 straight f left parenthesis x right parenthesis space straight d x equals 12 and thatspace integral subscript 10 superscript 15 straight f left parenthesis x right parenthesis space straight d x equals 5.

 

a)
Write down the values of
i)
space integral subscript 7 superscript 7 straight f left parenthesis x right parenthesis space straight d x
ii)
space integral subscript 10 superscript 5 straight f left parenthesis x right parenthesis space straight d x

 5-4-3-ib-sl-aa-only-we2-soltn-a

b)
Find the values of
i)
space integral subscript 5 superscript 15 straight f left parenthesis x right parenthesis space straight d x
ii)
space integral subscript 5 superscript 10 6 straight f left parenthesis x plus 5 right parenthesis space straight d x

5-4-3-ib-sl-aa-only-we2-soltn-b

 

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Paul

Author: Paul

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.