AQA A Level Maths: Pure复习笔记8.1.3 Definite Integration

Definite Integration

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 AS/A level textbooks/websites

8.1.3-Notes-fig1

  • a and b are called limits
    • a is the lower limit
    • b is the upper limit
  • f’(x) is the derivative of f(x)

What happened to c, the constant of integration?

8.1.3-Notes-fig2

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

How do I find a definite integral?

  • STEP 1: If not given a name, call the integral
    • 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

8.1.3-Notes-fig3

Using a calculator

  • Advanced scientific calculators can work out the values of definite integrals
  • The button will look similar to:

8.1.3-Notes-fig4

8.1.3-Notes-fig5

  •  (Note how the calculator did not return the exact value