Integration as the limit of a sum (OCR A Level Maths: Pure)

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Integration as the limit of a sum

Finding the area under a curve

  • Definite integration allows us to find the area under a curve

Notes def_int, AS & A Level Maths revision notes

  •  An estimate for the area under the curve is the sum of the rectangular areas 

Notes area_2_rects, AS & A Level Maths revision notes

  • If the number of rectangles increases and their width decreases, the estimate is more accurate

Notes area_4_rects, AS & A Level Maths revision notes

  • The sum of the rectangle areas will have a limit, however small they get
    • The sum will become closer and closer to the area under the curve
    • This is called the limit of the sum

What is integration as the limit of a sum?

Notes area_of_1, A Level & AS Level Pure Maths Revision Notes

  • The width of a rectangle can be considered as a small increase along the x-axis
  • This is denoted by δx
  • The height (length) will be the y-coordinate at x1 – ie f(x1) (rather than f(x1+δx))
  • If we use four of these small rectangles between a and b we get

Notes area_of_4, A Level & AS Level Pure Maths Revision Notes

  • As more rectangles are used …
    • δx gets smaller and smaller, ie δx → 0
    • n, the number of rectangles, gets bigger and bigger, ie n → ∞
    • … the sum of the area of the rectangles becomes closer to the area under the curve

    Notes area_tends_lim, A Level & AS Level Pure Maths Revision Notes 

  • This is the meaning of integration as the limit of a sum

How do questions use integration as the limit of a sum?

  • STEP 1        Recognise the notation
  • STEP 2        Convert to a definite integral
  • STEP 3        Find the value of the integral

Notes lim_int_eg, AS & A Level Maths revision notes

Worked example

Example quest, AS & A Level Maths revision notesExample soltn, A Level & AS Level Pure 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.