Definite Integrals (Cambridge (CIE) O Level Additional Maths): Revision Note

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 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

    • ${\int }_a^{b}k\mathrm{f}\left(x\right)\mathrm{d}x=k{\int }_a^b\mathrm{f}\left(x\right)\mathrm{d}x$ where$k$ is a constant

    • useful when fractional and/or negative values involved

  • integrating term by term

    • ${\int }_a^b\left[\mathrm{f}\right(x)+\mathrm{g}(x\left)\right]\mathrm{d}x={\int }_a^b\mathrm{f}\left(x\right)\mathrm{d}x+{\int }_a^b\mathrm{g}\left(x\right)\mathrm{d}x$ 

    • the above works for subtraction of terms/functions too

  • equal upper and lower limits

    • ${\int }_a^a\mathrm{f}\left(x\right)dx=0$ 

    • on evaluating, this would be a value subtracted from itself!

  • swapping limits gives the same, but negative, result

    • ${\int }_a^b\mathrm{f}\left(x\right)\mathrm{d}x=-{\int }_b^a\mathrm{f}\left(x\right)\mathrm{d}x$ 

    • compare 8 subtract 5 say, with 5 subtract 8 …

  • splitting the interval

    • ${\int }_a^b\mathrm{f}\left(x\right)\mathrm{d}x={\int }_a^c\mathrm{f}\left(x\right)\mathrm{d}x+{\int }_c^b\mathrm{f}\left(x\right)\mathrm{d}x$ where$a\le c\le b$

    • this is particularly useful for areas under multiple curves or areas under the$x$-axis