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ASMaths: PureAQARevision Notes8. Integration8.1 Integration8.1.1 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus (AQA AS Maths: Pure)

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Fundamental Theorem of Calculus

What is the fundamental theorem of calculus?

Notes fig1, AS & A Level Maths revision notes

 

  • 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 “+c” (plus c)?

  •  When differentiating y, constant terms ‘disappear’
    • for constants y = c, fraction numerator d y over denominator d x end fraction equals 0
    • graphs of constants are horizontal lines and so have gradient open parentheses fraction numerator d y over denominator d x end fraction close parentheses of 0
  • Integrating fraction numerator d y over denominator 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

Notes fig2, AS & A Level Maths revision notes

Notation

Notes fig3, AS & A Level Maths revision notes

  • integral is the sign for integration
  • 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”
  • dx means integrate with respect to x, any other letter is treated like a number (ie like a constant)

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.