Section 55 Recall from Section 51 All of our RAM techniques utilized rectangles t o approximate areas under curves Another geometric shape may do this job m ore efficiently Trapezoids ID: 551110
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Slide1
Trapezoidal Rule
Section 5.5Slide2
Recall, from Section 5.1…
All of our RAM techniques utilized
rectangles
to approximate areas under curves.
Another geometric shape may do this
job
m
ore efficiently
Trapezoids!!!Slide3
Partition a function into
n
subintervals of equal length
h
= (
b – a)/n over the interval [a, b].
Approximate the area using the trapezoids:Slide4
Things to notice:
This technique is algebraically equivalent to finding the
numerical average of LRAM and RRAM!!!Slide5
To approximate , use
where [
a
,
b
] is partitioned into
n
subintervals of equal length
h
= (
b
–
a
)/n
.The Trapezoidal RuleSlide6
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
”:Slide7
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.Slide8
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???Slide9
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
68
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:Slide10
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
68
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.Slide11
Applying the Trapezoidal Rule
Let’s work through #8 on p.295…
(a) Estimate for volume using Trapezoidal Rule:Slide12
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.