The document discusses approximations of areas under curves using the trapezoidal rule. It introduces the trapezoidal rule formula and shows how it can be used to approximate areas with multiple intervals by summing the areas of individual trapezoids. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the other y-values, divided by two. An example demonstrates applying the rule to approximate the area under a curve between 0 and 2 using 4 intervals.

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) ba
A  f a   f b 
2
a b x

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

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

12. 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  c  a  f a   2 f c   2 f d   f b 
2
In general;

13. 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  c  a  f a   2 f c   2 f d   f b 
2
In general; b
Area   f  x dx
a

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

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

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

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

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

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

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

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

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

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

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

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
where h 
n
n  number of intervals

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

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

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

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

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

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

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

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

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

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

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