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Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
bfafab
A
2
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafab
A
2
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafab
A
2
c
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafab
A
2
c
bfcfcb
cfafac
A
22
Approximations To Areas (1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafab
A
2
c
bfcfcb
cfafac
A
22
bfcfafac
22
y
x
y = f(x)
a b
y
x
y = f(x)
a b d c
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
In general;
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
b
a
dxxfAreaIn general;
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
b
a
dxxfArea
nothers yyyh
22
0
In general;
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
b
a
dxxfArea
nothers yyyh
22
0
s trapeziumofnumber
where
n
n
abh
In general;
y
x
y = f(x)
a b d c
bfdfdb
dfcfcd
cfafac
A
2
22
bfdfcfafac
222
NOTE: there is
always one more
function value
than interval
b
a
dxxfArea
nothers yyyh
22
0
s trapeziumofnumber
where
n
n
abh
In general;
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
2units 996.2
03229.17321.19365.1222
5.0
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
2units 996.2
03229.17321.19365.1222
5.0
πe exact valu
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimate tointervals 4 with Rule lTrapezoida theUse
2
12 xxxy
5.0
4
02
n
abh
2units 996.2
03229.17321.19365.1222
5.0
πe exact valu
%6.4
100142.3
996.2142.3error %
nothers yyyh
22
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
(2) Simpson’s Rule
(2) Simpson’s Rule
b
a
dxxfArea
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g. x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
2units 084.3
07321.123229.19365.1423
5.0
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh
243
0
intervals ofnumber
where
n
n
abh
e.g.
2units 084.3
07321.123229.19365.1423
5.0
nevenodd yyyyh
243
Area 0
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
%8.1
100142.3
084.3142.3error %
Alternative working out!!! (1) Trapezoidal Rule
Alternative working out!!! (1) Trapezoidal Rule
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
Alternative working out!!! (1) Trapezoidal Rule
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 2 2 2
2 2 1.9365 1.7321 1.3229 0Area 2 0
1 2 2 2 1
22.996 units
(2) Simpson’s Rule
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
(2) Simpson’s Rule
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
2 4 1.9365 1.3229 2 1.7321 0Area 2 0
1 4 2 4 1
23.084 units
(2) Simpson’s Rule
x 0 0.5 1 1.5 2
y 2 1.9365 1.7321 1.3229 0
1 1 4 2 4
2 4 1.9365 1.3229 2 1.7321 0Area 2 0
1 4 2 4 1
23.084 units
Exercise 11I; odds
Exercise 11J; evens
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