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

Definite Integrals

What is a definite integral?

• This is known as the Fundamental Theorem of Calculus

• a and b are called limits

  • a is the lower limit

  • b is the upper limit

• is the integrand

• is an antiderivative of

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

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

  • subtracting means the “+c”’s 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

STEP 2

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

STEP 3

Substitute the limits into the function and evaluate

Properties of Definite Integrals

Fundamental Theorem of Calculus

• Formally,

  •  is 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 involved

  • integrating term by term

    •  

    • the above works for subtraction of terms/functions too

  • equal upper and lower limits

    •  

    • on evaluating, this would be a value, subtract itself !

  • swapping limits gives the same, but negative, result

    •  

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

  • splitting the interval

    •  where

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

  • horizontal translations

    •  where is a constant

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