Definite Integrals (DP IB Analysis & Approaches (AA)): Revision Note

Definite Integrals

What is a definite integral?

• A definite integral is written in the form , where

  •  is the integrand (function to be integrated)

  • and are the integration limits

    • is the lower limit, and is the upper limit

    • These correspond to the lines and in the area under a curve

• According to the Fundamental Theorem of Calculus, if  is an antiderivative of, then

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

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

  • Then subtracting means the “”’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.

• 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. 

•  STEP 3

  Substitute the limits into the function and evaluate

  • E.g. 

Properties of Definite Integrals

Fundamental Theorem of Calculus

• According to the Fundamental Theorem of Calculus

• In that equation

  •  must be continuous in the interval

  • is an antiderivative of

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

  •  where is a constant

    • useful when fractional and/or negative values are involved

• Integrating term by term

  • 

• Equal upper and lower limits

  • 

    • Because

• Swapping limits gives the negative of the original result

  •  

    • Because

• Splitting the interval

  •  where

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

• Horizontal translations

  •  where is a constant

    • The graph of is a horizontal translation of the graph of
      ( translates left,  translates right)
      

Summary of properties of definite integrals: