A LevelMaths: Pure 1CIERevision Notes6. Integration6.1 Integration6.1.1 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus (CIE A Level Maths: Pure 1)

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4 August 2023

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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, $\frac{d y}{d x} = 0$
- graphs of constants are horizontal lines and so have gradient $(\frac{d y}{d x})$ of 0
Integrating $\frac{d y}{d x}$ , 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

$\int$ 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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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.