Definite Integrals (DP IB Analysis & Approaches (AA): 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.$I={\left(2\right)}^3-{\left(1\right)}^3=8-1=7$ 

Properties of definite integrals

Fundamental Theorem of Calculus

• According to the Fundamental Theorem of Calculus

${\int }_a^{b}f\left(x\right)dx={\left[F\right(x\left)\right]}_a^b=F\left(b\right)-F\left(a\right)$

• In that equation

  • $f\left(x\right)$ must be continuous in the interval$a\le x\le b$

  • $F\left(x\right)$is an antiderivative of$\mathrm{f}\left(x\right)$

What are the properties of definite integrals?

• Some of these have been encountered already (and some may seem obvious)

• Taking constant factors outside the integral

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

    • useful when fractional and/or negative values are involved

• Integrating term by term

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

• Equal upper and lower limits

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

    • Because $F\left(a\right)-F\left(a\right)=0$

• Swapping limits gives the negative of the original result

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

    • Because $F\left(a\right)-F\left(b\right)=-\left(F\left(b\right)-F\left(a\right)\right)$

• Splitting the interval

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

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

• Horizontal translations

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

    • The graph of$y=f(x+k)$ is a horizontal translation of the graph of$y=f\left(x\right)$
      ($k>0$ translates left, $k<0$ translates right)
      

Summary of properties of definite integrals: