Problem:
Using the TI-85, approximate the value of

using the Simpson's Rule with a partition of 150 points.


Visualization:

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  1. We enter the function on the calculator:

  2. Subdivide [-1, 1] into 150 subintervals of equal length.
  3. The length of each of these subintervals is
    which is (1 - (-1))/150 = 1/75.
  4. We determine the endpoints of these subintervals:

    ai = -1 + i (1/75) = -1 + i/75

  5. We enter the left-hand endpoints on the calculator and store them in the variable x:

  6. The calculator displays the list of left-hand endpoints:

  7. The calculator automatically stores the values of f(xi) in the list y1.
  8. We now take the sum of the list y1, multiply by Dx, and store the answer in the variable L:

  9. We enter the right-hand endpoints on the calculator and store them in the variable x:

  10. The calculator displays the list of left-hand endpoints:

  11. Again, the calculator automatically stores the values of f(xi) in the list y1.
  12. We now take the sum of the list y1, multiply by Dx, and store the answer in the variable R:

  13. We take the average of the sum of the left-hand approximation and the right-hand approximation to get the Trapezoidal Rule and we store this in T:

    (L+R)/2->T

    and get the answer 1.56989143142.
  14. We enter the midpoints on the calculator and store them in the variable x:

  15. The calculator displays the list of midpoints:

  16. Again, the calculator automatically stores the values of f(xi) in the list y1.
  17. We now take the sum of the list y1, multiply by Dx, and store the answer in the variable M:

    and get the midpoint approximation: 1.57106123569.
  18. To get Simpson's Rule, we take the weighted average:

    (T+2*M)/3

    which gives the answer: 1.57067129882.

    The correct answer is p/2 = 1.57079... .