5.4.3 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)?

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)