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