Chapter 4.6
NUMERICAL INTEGRATION
After you finish your
HOMEWORK you should be
able to…
• Approximate a definite integral using
the Trapezoidal Rule
• Approximate a definite integral using
Simpson’s Rule
• Analyze the approximate errors in
the Trapezoidal Rule and Simpson’s
Rule
I lived from 1642-1715. I developed
Calculus. I called Calculus “Fluents and
Fluxions”. I discovered the law of
gravity. I generalized the Binomial
Theorem. Who am I?
A. Fermat
B. Newton
C. Pythagoras
D. Pascal
The use of the clickers
makes the class more
interesting.
A. Strongly Agree
B. Agree
C. Neutral
D. Disagree
E. Strongly Disagree
The Tablet PC is an
effective lecture tool.
A. Strongly Agree
B. Agree
C. Neutral
D. Disagree
E. Strongly Disagree
The lecture worksheets
help me understand the
material.
A. Strongly Agree
B. Agree
C. Neutral
D. Disagree
E. Strongly Disagree
Calculus is my favorite
class!
A. Strongly Agree
B. Agree
C. Neutral
D. Disagree
E. Strongly Disagree
6
The Trapezoidal Rule
4
2
5
-2
10
The Trapezoidal Rule
Let f be continuous on  a, b . The
Trapezoidal
Rule for approximating
b
 f  x  dx is given by
a
ba
 f  x0   2 f  x1   2 f  x2  
2n
 2 f  xn 1   f  xn  
As n approaches infinity, this
approximation approaches
b
 f  x  dx
a
Simpson’s Rule
Let f be continuous on  a, b . Simpson’s
Rule
for approximating
b
 f  x  dx is given by
a
ba
 f  x0   4 f  x1   2 f  x2   4 f  x3    4 f  xn1   f  xn 
3n
As n approaches infinity, this b
approximation approaches  f  x  dx
a
1
0.8
0.6
0.4
0.2
0.5
-0.2
1
1.5
If f has a continuous second derivative on  a, b , then the
b
error E in approximation
 f  x  dx by the Trapezoidal Rule is
a
b  a 
E
12n
2
3
 max f   x   ,
a  x  b.
Moreover, if f has a continuous fourth derivative on  a, b ,
b
then the error E in approximating
 f  x  dx by Simpson’s Rule
a
is
b  a 
E
180n
4
5
 max f  4   x   ,


a  x  b.