Definite Integrals and the Fundamental Theorem of Calculus

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• Questions?
  • Problem set question 2, hand calculation & R
    
    

    > dx = 1/2000
    > x = seq( 0, 1, by=dx )
    > x[1]
    [1] 0
    > x[2001]
    [1] 1
    > x[2000]
    [1] 0.9995
    > x[1:2000]^3
    > sum( x[1:2000]^3*dx )

• Definite Integrals
  • Applications
    • Area under curve
      • NMR
      • Works by aligning atoms with magnetic field, then flipping some and letting them return to alignment; when they do so they emit energy characteristic of where they are relative to other atoms
        
      • See http://www.brynmawr.edu/chemistry/Chem/mnerzsto/The_Basics_Nuclear_Magnetic_Resonance%20_Spectroscopy_2.htm for more information
    • Average value of a function
      
  • Rules
    
• The Fundamental Theorem
  • Section 5.4
    • Mean value theorem for integrals
      • Average value calculated from definite integral is really taken on by f (if f is continuous)
      • For example x+1 from 0 and 1 has definite integral 3/2
      • So average value is 1/1 × 3/2 = 3/2
      • Does x+1 = 3/2 somewhere in [0,1]?
      • Yes, namely at x = 1/2
    • Fundamental theorem, parts 1 and 2
      
    • Net change theorem
• Next
  • Some examples of definite integrals
  • Integration by substitution

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