5.5
Trapezoidal Rule
Quick Review
Tell whether the curve is concave up or concave down on the given interval.
1. y  cos x on [-1,0]
2. y  x  3 x  6 on [8,17]
4
2
 x
3. y  sin   on [48 ,50 ]
2
4. y  e on [-5,5]
5. y  1/ x on [4, 8]
3x
6. y  csc x on  0, 
7. y  sin x - cos x on [1,2]
Quick Review Solutions
Tell whether the curve is concave up or concave down on the given interval.
1. y  cos x on [-1,0] concave down
2. y  x  3 x  6 on [8,17] concave up
4
2
 x
3. y  sin   on [48 ,50 ] concave down
2
4. y  e on [-5,5] concave up
5. y  1/ x on [4, 8] concave up
3x
6. y  csc x on  0,  concave up
7. y  sin x - cos x on [1,2] concave down
What you’ll learn about
Trapezoidal Approximations
 Other Algorithms
 Error Analysis

Essential Question
How do we find definite integrals that are not
best found by numerical approximations,
and rectangles are not always the most
efficient figures to use?
Trapezoidal Approximations
y y
y y
y y
 h
 ...  h 
2
2
2
y 
y
 h   y  y  ...  y  
2
2
h
  y  2 y  2 y  ...  2 y  y  ,
2
where y  f (a ), y  f ( x ), ..., y  f ( x ), y  f (b).
 f ( x) dx  h 
b
0
1
1
n 1
2
n
a
0
n
1
0
0
n 1
2
1
1
n 1
2
1
n 1
n
n 1
n
The Trapezoidal Rule
To approximate  f ( x)dx, use
h
T   y  2 y  2 y  ...  2 y  y  ,
2
where [a, b] is partitioned into n subintervals of equal length
h  (b - a ) / n.
LRAM  RRAM
Equivalently, T 
,
2
where LRAM and RRAM are the Rienamm sums using the left
b
a
0
1
n 1
2
n
n
n
n
n
and right endpoints, respectively, for f for the partition.
Example Using Trapezoidal Rule
1. Use the Trapezoidal Rule with n = 4 to approximate the value of
the integral. Use the concavity of the function to predict whether
the approximation is an overestimate or an underestimate.

4
0
x dx n  4
h 401
4
h
T   y0  2 y1  2 y2  2 y3  y4 
2
       4 
1
T  0  2 1  2 2  2 3 
2
 5.146
underestimate
Simpson’s Rule
To approximate  f ( x)dx, use
h
S   y  4 y  2 y  4 y  ...  2 y  4 y  y  ,
3
where [a, b] is partitioned into an even number n subintervals
of equal length h  (b - a ) / n.
b
a
0
1
2
3
n2
n 1
n
Example Using Simpson’s Rule
2. Use Simpson’s Rule with n = 4 to approximate the value of the
integral. Find the integral’s exact answer to check your answer.

2
0
x dx
1
2

0

h
4
2
n4
h
S   y0  4 y1  2 y2  4 y3  y4 
3
1
S  0  4.5  21  41.5  2   2
6
2
1 2 2 1
1 2
2
0 x dx  2 x 0   2 2    2 0   2


Error Bounds
If T and S represent the approximations to  f ( x)dx given by the
Trapezoidal Rule and Simpson's Rule, respectively, then the errors
E and E satisfy
b
a
T
s
b  ab  a
ba
E 
h M and
hM
2 E 
ES
ET 
h M f  and
12
180
2
T
4
f
12
n
s
f
(4)
ba 4

h M f 4 
180
Pg. 312, 5.5 #1-19 odd