Riemann Sums Project

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					Riemann Sums Project
○   Choose any interval [ a , b ].
○   Choose any function f (x) where f (x) > 0 for all x’s on the interval [ a , b ] .
○   Choose any number of partitions (greater than 5).
○   Your function on your chosen interval should be asymmetric.
For your function, interval, and number of partitions:

    (1) Find the Trapezoid sum. Illustrate and show all calculations.

    (2) Find the left Riemann sum. Illustrate and show all calculations.

    (3) Find the right Riemann sum. Illustrate and show all calculations.

    (4) Find the midpoint Riemann sum. Illustrate and show all calculations.

    (5) Find the exact definite integral. Illustrate and show all calculations.

    (6) Include a summary page of all five results and state whether the
        approximation overestimates or underestimates the exact integral.

    (7) All five illustrations should include an accurate graph. Use a straightedge to
       draw axes and partitions. The discrepancy between the rectangles/trapezoids
       and the actual curve should be discernable. Use color to emphasize exactly
       what area is being calculated.
How can you get a better estimate from a Riemann sum?

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What is the Trapezoid Rule for six partitions of uniform height h and bases
a, b1, b2, b3, b4, b5, b?

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Overestimate vs. Underestimate
Which of the following will overestimate and which will underestimate?
Illustrate the correct curve in the left margin.


Illustration   Slope and      Trapezoid     Left          Right          Midpoint
               Concavity      Rule          Riemann       Riemann        Riemann
                                            Sum           Sum            Sum

               Increasing &
               Concave Up



               Decreasing &
               Concave Up



               Increasing &
               Concave
               Down


               Decreasing &
               Concave
               Down

				
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posted:12/1/2011
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