Section 5.5
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].
Pn  xn , yn 
x0  a
xn  b
Approximate the area using the trapezoids:
P0  x0 , y0 

b
a
y0  y1
f  x  dx  h

2
yn 1  yn
h
2

b
a
y0  y1
yn 1  yn
y1  y2
f  x  dx  h
h
h
2
2
2
yn 
 y0
 h   y1  y2   yn 1  
2 
 2
h
  y0  2 y1  2 y2   2 yn 1  yn 
2
Things to notice:
y0  f  a  y1  f  x1  yn1  f  xn1  yn  f  b 
This technique is algebraically equivalent to finding the
numerical average of LRAM and RRAM!!!
The Trapezoidal Rule
 f  x  dx , use
b
To approximate
a
h
T   y0  2 y1  2 y2 
2
 2 yn 1  yn 
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.
2
2
Let’s start with a diagram…
1

Now, find “h”:
x dx
2 1 1
h

4
4
h
T   y0  2 y1  2 y2   2 yn 1  yn 
2
1
 25   36   49  
T  1  2    2    2    4 
8
 16   16   16  
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.
2
2

1
x dx
1
 25   36   49  
T  1  2    2    2    4 
8
 16   16   16  
75

 2.34375
32
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.
2
2

1
T  2.34375
x dx
Do we expect this to be an overestimate
or an underestimate? Why???
NINT  x , x,1, 2   2.33333333
2

2
1
2
x 
8 1 7
x dx     
3 1 3 3 3
3
2
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 1 2 3 4
63 65 66 68 70
5 6 7 8 9 10 11 M
69 68 68 65 64 62 58 55
What was the average temperature for the 12-hour period?
1 b
av  f  
f  x  dx

ba a
But we don’t have
a rule for f (x)!!!
We can estimate the area using the TR:
1
T   63  2 65  2 66 
2
 2 58  55 
 782
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 1 2 3 4
63 65 66 68 70
5 6 7 8 9 10 11 M
69 68 68 65 64 62 58 55
What was the average temperature for the 12-hour period?
1 b
av  f  
f  x  dx

ba a
1
1
av  f  
T 
782  65.167
ba
12
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:
200
V
0  2  520   2  800   2 1000  

2
2 1140   2 1160   2 1110  
2  860   0   20 
 26,360, 000ft
3
Applying the Trapezoidal Rule
Let’s work through #8 on p.295…
(b) You plan to start with
You intend to have
caught.
Since
26,360, 000
 26,360
1000
 0.75 26,360  19,770
fish.
fish to be
19, 770
 988.5 ,
20
the town can sell at most 988 licenses.