Numerical Integration

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• Misc
  • Exam 1
    • Next Tuesday (March 3)
    • Material since start of semester (e.g., ln and e^x, L’Hôpital’s Rule, inverse trig functions, hyperbolic functions, integration by parts, integration via trig substitution, partial fractions, numerical integration, improper integrals)
    • 4 - 6 short-answer questions, similar to problem set questions
    • Whole class period
    • Open book, notes, computer for reference; closed person, no CASes
  • Colloquium
    • “An Elementary Construction of Tessellated Surfaces”
    • Jonathan Pakianathan
    • University of Rochester
    • Monday, March 2, 4:00 - 5:00, Newton 201
    • Extra credit for summaries
• Questions?
• Integration Problem Set
  • Handout
• Numerical Integration
  • Section 8.7
    • Trapezoid rule similar to Reimann sums but w/ trapezoids rather than rectangles
    • Simpson’s rule - Parabolas?
      
    • Error analysis
      • Trapezoids: 2nd derivative, and n²
        • E = M(b-a)³/kn²
      • Simpson: 4th derivative and n⁴
        • E = M(b-a)⁵ / cn⁴
  • Example: integrate y = sinx between 0 and π using 4 intervals, using Simpson’s rule
    
    • R commands to do this calculation and estimate error
      

      > xs = c( 0, pi/4, pi/2, 3*pi/4, pi )
      > ys = sin( xs )
      > coeffs = c( 1, 4, 2, 4, 1 )
      > ys * coeffs
      [1] 0.000000e+00 2.828427e+00 2.000000e+00 2.828427e+00 1.224647e-16
      > sum( ys * coeffs )
      [1] 7.656854
      > (pi/12) * sum( ys * coeffs )
      [1] 2.00456
      > M = 1
      > M * (pi)^5 / (180*4^4)
      [1] 0.006641052

• Next
  • Improper integrals
  • Read section 8.8

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