Definite Integrals (DP IB Applications & Interpretation (AI): HL): Revision Note

Definite integrals

What is a definite integral?

• A definite integral is written in the form ${\int }_a^{b}f\left(x\right)\mathrm{d}x$, where

  • $f\left(x\right)$ is the integrand (function to be integrated)

  • $a$ and $b$ are the integration limits

    • $a$ is the lower limit, and $b$ is the upper limit

    • These correspond to the lines $x=a$ and $x=b$ in the area under a curve

• According to the Fundamental Theorem of Calculus, if $F\left(x\right)$ is an antiderivative of$f\left(x\right)$, then

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

• The constant of integration (“$+c$”) is not needed in definite integration

  • "$+c$” would appear alongside both F(a) and F(b)

  • Then subtracting means the “$+c$”’s would cancel

How do I find definite integrals analytically (manually)?

• STEP 1

  Give the integral a name to save having to rewrite the whole integral every time

  If need be, rewrite the integral into an integrable form

  • E.g.$I={\int }_1^{2}3x^2\mathrm{d}x$

• STEP 2

  Integrate without applying the limits; you will not need “+c”
  Notation: use square brackets [ ] with limits placed at the end bracket

  • E.g.$I={\left[x^3\right]}_1^2$ 

•  STEP 3

  Substitute the limits into the function and evaluate

  • E.g.$I={\left(2\right)}^3-{\left(1\right)}^3=8-1=7$