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Section 5.5

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

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

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

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

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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”:

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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.

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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???

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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:

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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.

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

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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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