STATEMENT OF PROBLEM

Below is the graph of an irregularly shaped closed region drawn to scale.

1. Make a very quick - almost instantaneous reaction - estimate of its area.

2. Take a moment, no more than 1 minute, to assess the precision of your answer to Problem 1. Was you initial estimate good or bad. Why? Can you improve your estimate? Why do you feel your second estimate is an improvement?

3. Make a table of values for the vertical distance between the top and bottom of the region at x = 0, 1, 2, . . . ,10. Estimate as necessary.

4. Compare Numerical Integration Techniques.

a) Explain why the midpoint rule is inappropriate to use in this problem.

b) Use your data from Problem 3 and the trapezoid rule to estimate the area of the region.

c) Use your data from Problem 3 and Simpson's rule to estimate the area of the region.

5. Expand the table from problem 3 to account for vertical distances at
x = 0, .5, 1, 1.5, . . . , 9.5, 10.

6. Repeat Problem 4 with the expanded table from Problem 5.

7. Compare your answers from Problem 2, Problem 4 and Problem 6 and discuss your results. In which estimate(s) do you have the most faith? Why?

8. The region we are studying has been determined by 5 functions, each defined over a different domain. Ask your teacher for the functions, and then compute the exact area using definite integrals.

9. Make a table listing the percent error in each of the estimates made: (in Problem 1, Problem 2, Problem 4 and Problem 6.

% relative error = ((approx value - exact value)/exact value) x 100%.
This is a signed error. If the error is positive, then the approximation is too big, and if the error is negative, the approximation is too small.

10. Evaluate this exercise. Did you like it? Why or why not? How could it be improved? Why do you think the actual functions were withheld until Problem 8?