Introduction to Numerical Integration

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• Misc
  • 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
  • Exam 1 is Monday (March 2)
    • Material since beginning of semester (e.g., expressions, variables, vectors, matrices, plotting, scripts, functions, etc.)
    • 3 - 5 short-answer questions
      • Old exam for examples
    • May ask you to write fragments of Matlab code, but syntax details won’t count for much in grading
    • Whole class period
    • Open book, notes, computer for reference; closed person
  • I’m out of town Friday March 6
    • Class will be review of test solutions with Nick
• Questions?
• Numerical Integration Algorithms
  • Riemann sum videos
    • Rectangles under curve, heights go up to curve, width is change in x
    • Add up values of function times width, i.e., area of rectangle
    • Can use either left or right sides of rectangles as x values
      • Over- or under-estimate integral depending on concavity of curve
    • Limit as number of rectangles goes to infinity (i.e., as width of rectangles goes to 0) is definition of the definite integral
  • Trapezoid rule video
    • Like Riemann but use trapezoids
    • Each side reaches function, like left and right Riemann at same time
    • Value of integral is sum of areas
      • = Width of trapezoids times sum of first function value plus last plus twice every in-between value
  • Next lab (after exam) will be to approximate definite integrals as Riemann sums or sums of trapezoids and plot functions and rectangles/trapezoids
  • Riemann sum algorithm
    • Algorithm needs to describe
      • Inputs
      • Series of calculations (i.e., steps)
      • Outputs
    • Example(s) of problem
      • y = f(x) = x between 0 and 3 with 3 rectangles
      • Inputs
        • Range of x values
        • # of rectangles
• Next
  • (After exam)
  • Look at steps for calculating Riemann sum

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