Introduction to Integration (Cambridge O Level Additional Maths)

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Introduction to Integration

What is integration?

  • Integration is the opposite to differentiation
  • Integration is the process of finding the expression of a function from an expression of the derivative (gradient function)

What is the fundamental theorem of calculus?

Integration reverses differentiation

 

  • The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation
    • This form of the Theorem relates to Indefinite Integration
    • An alternative version of the Fundamental Theorem of Calculus involves Definite Integration

What is the constant of integration (+c)?

  •  When differentiating y, constant terms ‘disappear’
    • for constants y = c, fraction numerator straight d y over denominator straight d x end fraction equals 0
    • graphs of constants are horizontal lines and so have gradient of 0
  • Integrating fraction numerator straight d y over denominator straight d x end fraction, to get y, cannot determine the constant
    • To acknowledge this constant, “+ c” is used
    • c is called the constant of integration

Example of a family of functions that differentiate to the same function

What is the notation for integration?

  • An integral is normally written in the form

integral f left parenthesis x right parenthesis space straight d x 

    • the large operator integral means “integrate”
    • straight d x” indicates which variable to integrate with respect to
    • space f left parenthesis x right parenthesis is the function to be integrated
  • If it has more than one term the function to be integrated (called the integrand) should be in brackets
    • “Integrate” -–  “all of (…)”  -–  “with respect to x”

Example of the notation for integration

Worked example

Example fig1, AS & A Level Maths revision notes

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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.