Warm-Up: February 13, 2013

1. Warm-Up: February 13, 2013 • Use your RIEMANN1 program to estimate • Find each of the following: • Left sum = • Right sum = • Trapezoid sum =

2. Homework Questions

3. Trapezoidal Rule Section 5.5

4. Trapezoidal Approximations • Instead of using rectangles to approximate the area under a curve, we can use trapezoids. • Partition [a, b] into n subintervals of equal length. This is the height of each trapezoid. • The trapezoidal approximation of is

5. Example 1 • Use the Trapezoidal Rule with n=4 to approximate 1.89612

6. Problems with Trapezoids • Uses straight lines to approximate curves. • Overestimates if the graph is concave up. • Underestimates if the graph is concave down.

7. Estimating with Parabolic Curves • Instead of using straight lines to represent curves, we can use parabolic arcs. • The area under a parabolic arc is

8. Simpson’s Rule • Partition [a, b] into an even number of subintervals, n. • Simpson’s Rule approximates

9. Example 2 • Use Simpons’s Rule with n=4 to approximate 2.004560

10. Assignment • Read Section 5.5 (pages 289-294) • Page 295 Exercises 1-9 odd, 13, 15 • For 13 and15, do n=4 and n=6

12. Warm-Up: February 14, 2013 • Use Simpons’s Rule with n=4 to approximate 2.004560

13. Homework Questions?

14. Error Analysis • Trapezoidal Rule and Simpson’s Rule are approximations. • Since they do not give the exact answer, they have some error. • What does the amount of error depend on?

15. Error Bounds • ET is the error of the Trapezoidal Rule • ES is the error of Simpson’s Rule • h is the length of each subinterval • Mf’’ is the maximum value of |f’’| • Mf(4) is the maximum value of |f(4)|

16. Assignments • Read Section 5.5 (pages 289-294) • Page 295 Exercises 1-9 odd, 13, 15 • For 13 and 15, do n=4 and n=6 • Page 296 #10 • Use complete sentences (plural) for part d • Page 298 #1-57 odd (Review Assignment)