Definite Integrals (Cambridge O Level Additional Maths)

What is definite integration?

• Definite Integration occurs in an alternative version of the Fundamental Theorem of Calculus
• This version of the Theorem is the one referred to by most textbooks/websites

• a and b are called limits
  • a is the lower limit
  • b is the upper limit
• f’(x) is the derivative of f(x)
• The value can be positive, zero or negative

Why do I not need to include a constant of integration for definite integration?

•  “+c” would appear in both f(a) and f(b)
  • Since we then calculate f(b) – f(a) they cancel each other out
  • So “+c” is not included with definite integration

How do I find a definite integral?

• STEP 1
  • Give the integral a name (if it does not already have one) 
    • This saves you having to rewrite the whole integral every time!
• STEP 2
  • If necessary rewrite the integral into a more easily integrable form
    • Not all functions can be integrated directly
• STEP 3
  • Integrate without applying the limits
    • Notation: use square brackets [ ] with limits placed after the end bracket
• STEP 4
  • Substitute the limits into the function and calculate the answer
    • Substitute the top limit first
    • Then substitute the bottom limit
    • Subtract the second value from the first

What are the special properties of definite integrals?

• Some of these have been encountered already and some may seem obvious …
  • taking constant factors outside the integral
    •  where is a constant
    • useful when fractional and/or negative values involved
  • integrating term by term
    •  
    • the above works for subtraction of terms/functions too
  • equal upper and lower limits
    •  
    • on evaluating, this would be a value subtracted from itself!
  • swapping limits gives the same, but negative, result
    •  
    • compare 8 subtract 5 say, with 5 subtract 8 …
  • splitting the interval
    •  where
    • this is particularly useful for areas under multiple curves or areas under the-axis