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Section 5.5. All of our RAM techniques utilized rectangles to approximate areas under curves. Another geometric shape may do this job more efficiently.

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Presentation on theme: "Section 5.5. All of our RAM techniques utilized rectangles to approximate areas under curves. Another geometric shape may do this job more efficiently."— Presentation transcript:

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 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 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.


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