Definite Integrals (Cambridge (CIE) IGCSE Additional Maths) : Revision Note

Paul

Author

Paul

Last updated

Did this video help you?

Definite Integration

What is definite integration?

  • Definite Integration occurs in an alternative version of the Fundamental Theorem of Calculus

  • This version of the Theorem is the one referred to by most textbooks/websites

Fundamental Theorem of Calculus using definite integration
  • a and b are called limits

    • a is the lower limit

    • b is the upper limit

  • f’(x) is the derivative of f(x)

  • The value can be positive, zero or negative

Why do I not need to include a constant of integration for definite integration?

Example of the constant of integration cancelling out
  •  “+c” would appear in both f(a) and f(b)

    • Since we then calculate f(b)f(a) they cancel each other out

    • So “+c” is not included with definite integration

How do I find a definite integral?

  • STEP 1

    • Give the integral a name (if it does not already have one) 

      • This saves you having to rewrite the whole integral every time!

  • STEP 2

    • If necessary rewrite the integral into a more easily integrable form

      • Not all functions can be integrated directly

  • STEP 3

    • Integrate without applying the limits

      • Notation: use square brackets [ ] with limits placed after the end bracket

  • STEP 4

    • Substitute the limits into the function and calculate the answer

      • Substitute the top limit first

      • Then substitute the bottom limit

      • Subtract the second value from the first

Example of definite integration

What are the special 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 subtracted from 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

Examiner Tips and Tricks

  • 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)

Worked Example

Example fig1, A Level & AS Level Pure Maths Revision Notes
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

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.

Download notes on Definite Integrals