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Trapezoidal Rule

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Trapezoidal Rule

Section 5.5

Recall, from Section 5.1…

All of our RAM techniques utilized rectangles

to approximate areas under curves.

Another geometric shape may do this job

more efficiently Trapezoids!!!

Partition a function into n subintervals of equal length

h = (b – a)/n over the interval [a, b].

Approximate the area using the trapezoids:

Things to notice:

This technique is algebraically equivalent to finding the

numerical average of LRAM and RRAM!!!

The Trapezoidal Rule

To approximate , use

where [a, b] is partitioned into n subintervals of equal length

h = (b – a)/n.

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”:

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.

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???

Applying the Trapezoidal Rule

An observer measures the outside temperature every hour from

noon until midnight, recording the temperatures in the following

table.

Time

N

1

2

3

4

5

6

7

8

9

10

11

M

Temp

63

65

66

68

70

69

68

68

65

64

62

58

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:

Applying the Trapezoidal Rule

An observer measures the outside temperature every hour from

noon until midnight, recording the temperatures in the following

table.

Time

N

1

2

3

4

5

6

7

8

9

10

11

M

Temp

63

65

66

68

70

69

68

68

65

64

62

58

55

What was the average temperature for the 12-hour period?

We estimate the average temperature to be about 65 degrees.

Applying the Trapezoidal Rule

Let’s work through #8 on p.295…

(a) Estimate for volume using Trapezoidal Rule:

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.