11x1 t14 07 approximations
TRANSCRIPT
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
bfafabA
2
Approximations To Areas(1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafabA
2
Approximations To Areas(1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafabA
2
c
Approximations To Areas(1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafabA
2
c
bfcfcbcfafacA
22
Approximations To Areas(1) Trapezoidal Rule
y
x
y = f(x)
a b
y
x
y = f(x)
a b
bfafabA
2
c
bfcfcbcfafacA
22
bfcfafac
2
2
y
x
y = f(x)
a b
y
x
y = f(x)
a bdc
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
In general;
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
b
a
dxxfAreaIn general;
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
b
a
dxxfArea
nothers yyyh 2
2 0
In general;
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
b
a
dxxfArea
nothers yyyh 2
2 0
s trapeziumofnumber
where
nn
abh
In general;
y
x
y = f(x)
a bdc
bfdfdb
dfcfcdcfafacA
2
22
bfdfcfafac
22
2
NOTE: there is always one more function value than interval
b
a
dxxfArea
nothers yyyh 2
2 0
s trapeziumofnumber
where
nn
abh
In general;
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 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
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 12 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
2units996.2
03229.17321.19365.12225.0
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 12 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
2units996.2
03229.17321.19365.12225.0
πe exact valu
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 12 2 2
e.g.
points decimal 3 correct to
2 and 0between ,4 curve under the area
theestimatetointervals4with RulelTrapezoida theUse
21
2 xxxy
5.04
02
nabh
2units996.2
03229.17321.19365.12225.0
πe exact valu
%6.4
100142.3
996.2142.3error %
nothers yyyh 2
2Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 12 2 2
(2) Simpson’s Rule
(2) Simpson’s Rule
b
a
dxxfArea
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 1
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 14 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 14 2 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
2units 084.3
07321.123229.19365.14235.0
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 14 2 4
(2) Simpson’s Rule
b
a
dxxfArea
nevenodd yyyyh 24
3 0
intervalsofnumber
where
nn
abh
e.g.
2units 084.3
07321.123229.19365.14235.0
nevenodd yyyyh 24
3Area 0
x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0
1 14 2 4
%8.1
100142.3
084.3142.3error %
Exercise 11I; odds
Exercise 11J; evens