1. Section 5.5

2. All of our RAM techniques utilized rectangles to approximate areas under curves. Another geometric shape may do this job more efficiently  Trapezoids!!!

3. Partition a function into n subintervals of equal length h = (b – a)/n over the interval [a, b]. Approximate the area using the trapezoids:

4. Things to notice: This technique is algebraically equivalent to finding the numerical average of LRAM and RRAM!!!

5. To approximate, use where [a, b] is partitioned into n subintervals of equal length h = (b – a)/n. The Trapezoidal Rule

6. Applying the Trapezoidal Rule Use the Trapezoidal Rule with n = 4 to estimate the given integral. Compare the estimate with the NINT value and with the exact value. Let’s start with a diagram… Now, find “h”:

7. Applying the Trapezoidal Rule Use the Trapezoidal Rule with n = 4 to estimate the given integral. Compare the estimate with the NINT value and with the exact value.

8. Applying the Trapezoidal Rule Use the Trapezoidal Rule with n = 4 to estimate the given integral. Compare the estimate with the NINT value and with the exact value. Do we expect this to be an overestimate or an underestimate? Why???

9. Applying the Trapezoidal Rule An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table. Time Temp N 63 1 65 2 66 3 68 4 70 5 69 6 68 7 8 65 9 64 10 62 11 58 M 55 What was the average temperature for the 12-hour period? But we don’t have a rule for f (x)!!! We can estimate the area using the TR:

10. Applying the Trapezoidal Rule An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table. Time Temp N 63 1 65 2 66 3 68 4 70 5 69 6 68 7 8 65 9 64 10 62 11 58 M 55 What was the average temperature for the 12-hour period? We estimate the average temperature to be about 65 degrees.

11. Applying the Trapezoidal Rule Let’s work through #8 on p.295… (a) Estimate for volume using Trapezoidal Rule:

12. Applying the Trapezoidal Rule Let’s work through #8 on p.295… (b) You plan to start with fish. You intend to have fish to be caught. Since, the town can sell at most 988 licenses.