sigma notations example this tells us to start with k=1 this tells us to end with k=100 this tells...
TRANSCRIPT
Sigma Notations
Example
100
1
2
k
kThis tells us tostart with k=1
This tells us toend with k=100
This tells usto add.
3
1 1k k
k
Formula2
)1(
1
nni
n
i
6
)12)(1(
1
2
nnni
n
i
2
1
3
2
)1(
nni
n
i
nn
i
1
1
Formula2
)1(
1
nni
n
i
6
)12)(1(
1
2
nnni
n
i
2
1
3
2
)1(
nni
n
i
nn
i
1
1
Example
3
1
2 )34(i
ii
Sigma Notations
Riemann Sum
0xa nxb
1x 2x 3x
},,,,{ 210 nxxxxP is called a partition of [a, b].
Example
}10,8,6,4,2,0{P Is a partition of [0, 10].
}9,6,3,0{P Is a partition of [0, 9].
}10,8,7,4,0{P Is a partition of [0, 10].
Example }10,8,7,4,0{P Is a partition of [0, 10].
the largest of all the subinterval widths
1 kkk xxx subinterval widths
1x 3x
4x 2x
P P
Riemann Sum
Riemann sum for ƒ on the interval [a, b].
n
kkkp xcfS
1
)(
],[ lsubinterva in thechosen point a is 1 kkk xxc
2xy
],1,4
3[ ],
4
3,
2
1[ ],
2
1,
4
1[ ],
4
1,0[
Find Riemann sum
point endight rck
Example
Riemann Sum
Riemann sum for ƒ on the interval [a, b].
n
kkkp xcfS
1
)(
],[ lsubinterva in thechosen point a is 1 kkk xxc
2xy
],1,4
3[ ],
4
3,
2
1[ ],
2
1,
4
1[ ],
4
1,0[
Find Riemann sum
point endeft lck
Example
Riemann Sum
We start by subdividing the interval [a,b] into n subintervals
The width of the interval [a,b] is b-a
the width of each subinterval is n
abx
The subintervals are ],[, ],,[ ],,[ 12110 nn xxxxxx
ax 0
xax 1
xax 22
xax 33
xkaxk
bxn
n
kkn xxfR
1
)( point endRigth
n
kkn xxfL
11)( point endleft
Riemann Sum
n
abx
n
kkn xxfR
1
)(point endRigth
n
kkn xxfL
11)(
point endleft
Step
1
nxxx ,,,
21
Step
2 xkaxk
Step
3
point Mid
n
kkn xxfM
1
* )( xkxk )(2
1*
Riemann Sum
Riemann sum for ƒ on the interval [a, b].
n
kkkp xcfS
1
)(
],[ lsubinterva in thechosen point a is 1 kkk xxc
Find Riemann sum
point endight rck
Example
partition the interval [0,1] inton equal subintervals
Riemann Sum
Riemann sum for ƒ on the interval [a, b].
n
kkkp xcfS
1
)(
],[ lsubinterva in thechosen point a is 1 kkk xxc
Find Riemann sum
point endight rck
Example
partition the interval [0,1] inton equal subintervals
1
1)(
x
xf
Riemann Sum
Definition:
The Definite Integral
the definite integral of ƒ over [a, b] pP
S0
lim
n
kkk
nxcf
1
)(lim
Example
the definite integral of ƒ over [0, 1]
Find
Definition:
the definite integral of ƒ over [a, b] pP
S0
lim
n
kkk
nxcf
1
)(lim
Remark:
the definite integral of ƒ over [a, b] nnR
lim
nnL
lim n
nM
lim
The Definite Integral
Notation:
the definite integral of ƒ over [a, b] b
a
dxxf )(
Remark:n
nR
lim
b
a
dxxf )(
The Definite Integral
Remark:n
nR
lim
b
a
dxxf )(
The Definite Integral
nn
nn
LR
limlim
n
ii
nxxf
1
*)(lim
Area under
the curve
b
adxxf )(
the definite integral of f from a to b
nnM
lim
If you are asked to find one of
them choose the easiest one.
The Definite Integral
Example:
Example:Evaluate the following integrals by interpreting each in terms of areas.
1
0
21) dxxa
3
0)1() dxxb
Evaluate the following integrals by interpreting each in terms of areas.
The Definite Integral
THE DEFINITE INTEGRAL
Term-103
Property (1)
THE DEFINITE INTEGRAL
b
a
a
bdxxfdxxf )()(
Example:
0 2 cos
dxxx
0
2 cos dxxx
THE DEFINITE INTEGRAL
Property (2)
0)( a
adxxf
THE DEFINITE INTEGRAL
Property (3)
b
c
c
a
b
adxxfdxxfdxxf )()()(
THE DEFINITE INTEGRAL
Term-091
EXAM-1TERM-102
The Definite Integral
THE DEFINITE INTEGRAL
Term-092
THE DEFINITE INTEGRAL
THE DEFINITE INTEGRAL
Term-092
THE DEFINITE INTEGRAL
Term-082
Term-092
Term-082
THE DEFINITE INTEGRAL
Term-103
DEFINITION )(xf ],[ ba
],[ interval on the of valueaverage baffavg
b
aavg dxxfab
f )(1
meanfavg
