8.2.1 Integration as the limit of a sum

Finding the area under a curve

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

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

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

• 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?

• 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

• 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

   

• 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