MRS EZRINDA MOHD ZAIHIDEE

1
CHAPTER 5 NUMERICAL INTEGRATION
MRS EZRINDA MOHD ZAIHIDEE FAKULTI KEJ. ELEKTRIK
ELEKTRONIK
2
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
2
3
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
3
4
5.1 INTRODUCTION
Area of the shaded region?
4
5
5.1 INTRODUCTION
Method 1
Analytical
Method 2
Numerical
5
6
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
6
7
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• derive the formula of Trapezium rule by refer to
  the graph.
• solve the given questions using the formula.

7
8
5.2.1 FORMULA OF TRAPEZIUM RULE
y
f(x)
x
xnb
0
x0a
x1
x2
x3
x4
x5
8
9
5.2.1 FORMULA OF TRAPEZIUM RULE
The area under the curve sum of areas of n
trapezium
a
b
h
9
10
5.2.1 FORMULA OF TRAPEZIUM RULE
10
11
5.2.1 FORMULA OF TRAPEZIUM RULE
EXAMPLE 1 Evaluate using
trapezium rule with 5 strips.
11
12
5.2.1 FORMULA OF TRAPEZIUM RULE

• EXAMPLE 2
• Use the trapezium rule to estimate
• using
• a strip width of 0.2
• a strip width of 0.1

12
13
5.2.1 FORMULA OF TRAPEZIUM RULE
EXAMPLE 3 Use the trapezium rule to estimate the
value of with
13
14
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
14
15
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• calculate the maximum error using upper bound for
  trapezium rule.
• solve all the given question.

15
16
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
Error ? the difference between the estimated
value of the integral and the true
value of the integral
16
17
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
To find the maximum value of error (upper bound
for the error), suppose that
Then,
17
18
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
EXAMPLE 4 Find an upper bound for the error in
the estimates calculated in example 2. Hence,
find upper and lower bounds for the true value of
18
19
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
EXAMPLE 5 Find an upper bound for the error in
the estimates calculated in example 3. Hence,
find upper and lower bounds for the true value of
19
20
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
20
21
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• relate the formula of Simpsons rule by refer to
  the graph.
• solve the given questions using the formula.

21
22
5.3.1 FORMULA OF SIMPSONS RULE
y
D
E
C
F
B
A
f(x)
x
xa
xb
0
22
23
5.3.1 FORMULA OF SIMPSONS RULE
23
24
5.3.1 FORMULA OF SIMPSONS RULE

• EXAMPLE 6
• Use the Simpsons rule to estimate
• with
• 4 strips
• 8 strips

24
25
5.3.1 FORMULA OF SIMPSONS RULE
EXAMPLE 7 Estimate using
Simpsons rule with
25
26
5.2.1 FORMULA OF TRAPEZIUM RULE
5.3.1 FORMULA OF SIMPSONS RULE
5.3.2 ERROR DUE TO THE SIMPSONS RULE
5.2.2 ERROR DUE TO THE TRAPEZIUM RULE
26
27
LEARNING OUTCOMES

• By the end of this lecture, students should be
  able to
• calculate the maximum error using upper bound for
  Simpsons rule.
• solve all the given question.

27
28
5.3.2 ERROR DUE TO THE SIMPSONS RULE
Error ? the difference between the estimated
value of the integral and the true
value of the integral
28
29
5.3.2 ERROR DUE TO THE SIMPSONS RULE
To find the maximum value of error (upper bound
for the error), suppose that
Then,
29
30
5.3.2 ERROR DUE TO THE SIMPSONS RULE
EXAMPLE 8 Find an upper bound for the error in
the estimates calculated in example 6. Hence,
find upper and lower bounds for the true value of
30
31
5.3.2 ERROR DUE TO THE SIMPSONS RULE
EXAMPLE 9 Find an upper bound for the error in
the estimates calculated in example 7. Hence,
find upper and lower bounds for the true value of
31
32
EXERCISES

• 1. Use the trapezium rule and Simpsons rule to
  estimate
• (a)
• (b)
• Find an upper bound for the error in the
  estimates calculated in question 1(a) for
  trapezium rule. Hence, find upper and lower
  bounds for the true value of
• 3. Find an upper bound for the error in the
  estimates calculated in question 1(b) for
  Simpsons rule. Hence, find upper and lower
  bounds for the true value of

32