1. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
a
b
x

2. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
a
b
x

3. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
a
b
x
ba
A
 f a   f b 
2

4. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
ba
A
 f a   f b 
2
y
a
b
y = f(x)
x
a
b
x

5. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
ba
A
 f a   f b 
2
y
a
b
y = f(x)
x
a
c
b
x

6. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
ba
A
 f a   f b 
2
y
a
b
y = f(x)
x
ca
bc
A
 f a   f c  
 f c   f b 
2
2
a
c
b
x

7. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
ba
A
 f a   f b 
2
y
a
b
y = f(x)
x
ca
bc
A
 f a   f c  
 f c   f b 
2
2
ca

 f a   2 f c   f b 
2
a
c
b
x

8. y
y = f(x)
a
b
x

9. y
y = f(x)
a
c
d
b
x

10. y
y = f(x)
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
a
c
d
b
x

11. y
y = f(x)
a
c
d
b
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2

12. y
y = f(x)
a
c
In general;
d
b
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2

13. y
y = f(x)
a
c
In general;
d
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2
b
b
Area   f  x dx
a

14. y
y = f(x)
a
c
In general;
d
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2
b
b
Area   f  x dx
a
h
 y0  2 yothers  yn 
2

15. y
y = f(x)
a
c
In general;
d
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2
b
b
Area   f  x dx
a
h
 y0  2 yothers  yn 
2
ba
n
n  number of trapeziums
where h 

16. y
y = f(x)
a
c
In general;
d
ca
d c
A
 f a   f c  
 f c   f d 
2
2
bd

 f d   f b 
2
x  c  a  f a   2 f c   2 f d   f b 
2
b
b
Area   f  x dx
a
h
 y0  2 yothers  yn 
2
ba
n
n  number of trapeziums
where h 
NOTE: there is
always one more
function value
than interval

17. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
 , between x  0 and x  2
1
2 2

18. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
 , between x  0 and x  2
1
2 2

19. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
x
y
0
2
 , between x  0 and x  2
1
2 2
0.5
1.9365
1
1.7321
1.5
1.3229
2
0

20. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
x
y
0
2
 , between x  0 and x  2
1
2 2
0.5
1.9365
1
1.7321
h
Area  y0  2 yothers  yn 
2
1.5
1.3229
2
0

21. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
 , between x  0 and x  2
1
2 2
1
x
y
0
2
1
0.5
1.9365
1
1.7321
h
Area  y0  2 yothers  yn 
2
1.5
1.3229
2
0

22. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
1
x
y
0
2
 , between x  0 and x  2
1
2 2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  2 yothers  yn 
2

23. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
1
x
y
0
2
 , between x  0 and x  2
1
2 2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  2 yothers  yn 
2
0.5

2  21.9365  1.7321  1.3229  0
2
 2.996 units 2

24. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
1
x
y
0
2
 , between x  0 and x  2
1
2 2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  2 yothers  yn 
2
0.5

2  21.9365  1.7321  1.3229  0
2
exact value  π 
 2.996 units 2

25. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the
area under the curve y  4  x
correct to 3 decimal points 
ba
h
n
20

4
 0.5
1
x
y
0
2
 , between x  0 and x  2
1
2 2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  2 yothers  yn 
2
0.5

2  21.9365  1.7321  1.3229  0
2
exact value  π 
 2.996 units 2
3.142  2.996
100
3.142
 4.6%
% error 

26. (2) Simpson’s Rule

27. (2) Simpson’s Rule
b
Area   f  x dx
a

28. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3

29. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 

30. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
e.g.
x
y
0
2
0.5
1.9365
1
1.7321
1.5
1.3229
2
0

31. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
e.g.
x
y
0
2
0.5
1.9365
1
1.7321
h
Area  y0  4 yodd  2 yeven  yn 
3
1.5
1.3229
2
0

32. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
1
e.g.
x
y
0
2
1
0.5
1.9365
1
1.7321
h
Area  y0  4 yodd  2 yeven  yn 
3
1.5
1.3229
2
0

33. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
1
e.g.
x
y
4
0
2
0.5
1.9365
4
1
1.7321
h
Area  y0  4 yodd  2 yeven  yn 
3
1
1.5
1.3229
2
0

34. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
1
e.g.
x
y
4
2
4
1
0
2
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  4 yodd  2 yeven  yn 
3

35. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
1
e.g.
x
y
4
2
4
1
0
2
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  4 yodd  2 yeven  yn 
3
0.5

2  41.9365  1.3229  21.7321  0
3
 3.084 units 2

36. (2) Simpson’s Rule
b
Area   f  x dx
a
h
 y0  4 yodd  2 yeven  yn 
3
ba
n
n  number of intervals
where h 
1
e.g.
x
y
4
2
4
1
0
2
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
h
Area  y0  4 yodd  2 yeven  yn 
3
0.5

2  41.9365  1.3229  21.7321  0 3.142  3.084
3
% error 
100
3.142
 3.084 units 2
 1.8%

37. Alternative working out!!!
(1) Trapezoidal Rule

38. Alternative working out!!!
(1) Trapezoidal Rule
1
x
y
0
2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0

39. Alternative working out!!!
(1) Trapezoidal Rule
1
x
y
Area 
0
2
2
2
2
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
2  2 1.9365  1.7321  1.3229   0
1 2  2  2 1
 2.996 units 2
  2  0

40. (2) Simpson’s Rule
1
x
y
0
2
4
2
4
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0

41. (2) Simpson’s Rule
1
x
y
Area 
0
2
4
2
4
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
2  4 1.9365  1.3229   2 1.7321  0
1 4  2  4 1
 3.084 units 2
  2  0

42. (2) Simpson’s Rule
1
x
y
Area 
0
2
4
2
4
1
0.5
1.9365
1
1.7321
1.5
1.3229
2
0
2  4 1.9365  1.3229   2 1.7321  0
1 4  2  4 1
 3.084 units 2
Exercise 11I; odds
Exercise 11J; evens
  2  0