AQA A Level Maths: Pure复习笔记8.2.1 Integration as the limit of a sum

Integration as the limit of a sum

Finding the area under a curve

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

8.2.1-Notes-def_int

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

8.2.1-Notes-area_2_rects

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

8.2.1-Notes-area_4_rects

  • 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?

8.2.1-Notes-area_of_1

  • 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

8.2.1-Notes-area_of_4

  • 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

    8.2.1-Notes-area_tends_lim

  • 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

8.2.1-Notes-lim_int_eg

Worked Example

8.2.1-Example-quest8.2.1-Example-soltn

 

 

 

 

 

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