6. Approximations To Areas
(1) Trapezoidal Rule
y
y = f(x)
ba
A
f a f b
2
y
a
b
y = f(x)
x
ca
bc
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)
ba
A
f a f b
2
y
a
b
y = f(x)
x
ca
bc
A
f a f c
f c f b
2
2
ca
f a 2 f c f b
2
a
c
b
x
10. y
y = f(x)
ca
d c
A
f a f c
f c f d
2
2
bd
f d f b
2
a
c
d
b
x
11. y
y = f(x)
a
c
d
b
ca
d c
A
f a f c
f c f d
2
2
bd
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
ca
d c
A
f a f c
f c f d
2
2
bd
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
ca
d c
A
f a f c
f c f d
2
2
bd
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
ca
d c
A
f a f c
f c f d
2
2
bd
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
ca
d c
A
f a f c
f c f d
2
2
bd
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
ba
n
n number of trapeziums
where h
16. y
y = f(x)
a
c
In general;
d
ca
d c
A
f a f c
f c f d
2
2
bd
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
ba
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
ba
h
n
20
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
ba
h
n
20
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
ba
h
n
20
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
ba
h
n
20
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
ba
h
n
20
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
ba
h
n
20
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 21.9365 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
ba
h
n
20
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 21.9365 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
ba
h
n
20
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 21.9365 1.7321 1.3229 0
2
exact value π
2.996 units 2
3.142 2.996
100
3.142
4.6%
% error
29. (2) Simpson’s Rule
b
Area f x dx
a
h
y0 4 yodd 2 yeven yn
3
ba
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
ba
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
ba
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
ba
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
ba
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
ba
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
ba
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 41.9365 1.3229 21.7321 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
ba
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 41.9365 1.3229 21.7321 0 3.142 3.084
3
% error
100
3.142
3.084 units 2
1.8%