Integration as Area Sums

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• Misc
  • Make appointments for this week soon
  • Hour exam 2
    • Friday (Nov. 14)
    • Covers material from 1st exam through optimization (e.g., chain rule, implicit differentiation, related rates, Mean Value Theorem, curve sketching, extrema and optimization, etc.)
    • 7 - 9 short answer questions, including word problems
    • Rules and format otherwise similar to 1st exam, especially open references rule
• Questions?
  • Starting problem set question 2?
    • Start with 1st and 2nd derivatives, vertical asymptotes
  • Question 3
    
• Antiderivative Rules
  
• Integration and Sums
  • Section 5.1
    • Area under curve
      • Estimate from upper and lower sums
      • Midpoint rule?
    • Distance traveled or displacement?
    • Average value of function
    • Uses upper and lower sums for all of these
      
  • Can’t you do this with something like triangles too?
    
  • Estimating antiderivatives
    
    • Example
      • f′(x) = x²
      • Estimate f(2)
      • 4 intervals
        
      • Estimated f(2) = 3.75
      • Actual f(2) = 2.67
      • Problem is only that small number of intervals isn’t very accurate. Use R to calculate for a useful number of intervals:
        

        > x = seq(0,2,length=5)
        > x
        [1] 0.0 0.5 1.0 1.5 2.0
        > y = x^2
        > y
        [1] 0.00 0.25 1.00 2.25 4.00
        > areas = y * 0.5
        > sum(areas)
        [1] 3.75
        > x = seq(0,2,length=5000)
        > y = x^2
        > width = 2/4999
        > sum( y * width )
        [1] 2.667467

• Next
  • Summation notation, limits, and Riemann sums
  • Read section 5.2

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