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![Page 1: Summation Notation Also called sigma notationAlso called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series 2+4+6+8+10 can be written.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649ee15503460f94bf2526/html5/thumbnails/1.jpg)
Summation NotationSummation Notation• Also called Also called sigma notationsigma notation
(sigma is a Greek letter (sigma is a Greek letter ΣΣ meaning “sum”) meaning “sum”)The series 2+4+6+8+10 can be written as:The series 2+4+6+8+10 can be written as:
i is called the i is called the index of summationindex of summation Sometimes you will see an n or k here instead Sometimes you will see an n or k here instead
of i.of i.The notation is read:The notation is read:
““the sum from i=1 to 5 of 2i”the sum from i=1 to 5 of 2i”
5
1
2ii goes from 1 i goes from 1
to 5.to 5.
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Summation Notation for an Summation Notation for an Infinite SeriesInfinite Series
• Summation notation for the infinite series:Summation notation for the infinite series:
2+4+6+8+10+… would be written as:2+4+6+8+10+… would be written as:
Because the series is infinite, you must use i Because the series is infinite, you must use i from 1 to infinity (from 1 to infinity (∞) instead of stopping at ∞) instead of stopping at
the 5the 5thth term like before. term like before.
1
2i
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Examples: Write each series in Examples: Write each series in summation notation.summation notation.
a. 4+8+12+…+100a. 4+8+12+…+100• Notice the series can Notice the series can
be written as:be written as:
4(1)+4(2)+4(3)+…+4(25)4(1)+4(2)+4(3)+…+4(25)
Or 4(i) where i goes Or 4(i) where i goes from 1 to 25.from 1 to 25.
• Notice the series Notice the series can be written as:can be written as:
25
1
4i
...5
4
4
3
3
2
2
1 . b
...14
4
13
3
12
2
11
1
. to1 from goes where1
Or,
ii
i
1 1i
i
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ExampleExample: Find the sum of the : Find the sum of the series.series.
• k goes from 5 to 10.k goes from 5 to 10.
• (5(522+1)+(6+1)+(622+1)+(7+1)+(722+1)+(8+1)+(822+1)+(9+1)+(922+1)+(10+1)+(1022+1)+1)
= 26+37+50+65+82+101= 26+37+50+65+82+101
= = 361361
10
5
2 1k
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Estimating with Finite Sums
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
Greenfield Village, Michigan
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time
velocity
After 4 seconds, the object has gone 12 feet.
Consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
ft3 4 sec 12 ft
sec
3t d
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If the velocity is not constant,we might guess that the distance traveled is still equalto the area under the curve.
(The units work out.)
211
8V t Example:
We could estimate the area under the curve by drawing rectangles touching at their left corners.
This is called the Left-hand Rectangular Approximation Method (LRAM).
1 11
8
11
2
12
8t v
10
1 11
8
2 11
2
3 12
8Approximate area: 1 1 1 3
1 1 1 2 5 5.758 2 8 4
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We could also use a Right-hand Rectangular Approximation Method (RRAM).
11
8
11
2
12
8
Approximate area: 1 1 1 31 1 2 3 7 7.75
8 2 8 4
3
211
8V t
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Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).
1.031251.28125
1.78125
Approximate area:6.625
2.53125
t v
1.031250.5
1.5 1.28125
2.5 1.78125
3.5 2.53125
In this example there are four subintervals.As the number of subintervals increases, so does the accuracy.
211
8V t
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211
8V t
Approximate area:6.65624
t v
1.007810.25
0.75 1.07031
1.25 1.19531
1.382811.75
2.25
2.75
3.25
3.75
1.63281
1.94531
2.32031
2.75781
13.31248 0.5 6.65624
width of subinterval
With 8 subintervals:
The exact answer for thisproblem is .6.6
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When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.
211
8V t
subinterval
The width of a rectangle is called a subinterval.
1
limn
kn
k
f c x
If we let n = number of subintervals, then
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1
limn
kn
k
f c x
Leibnitz introduced a simpler notation for the definite integral:
1
limn b
k ank
f c x f x dx
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b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrandvariable of integration
(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
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time
velocity
After 4 seconds, the object has gone 12 feet.
Earlier, we considered an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance: 3t d
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
ft3 4 sec 12 ft
sec
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If the velocity varies:
11
2v t
Distance:21
4s t t
(C=0 since s=0 at t=0)
After 4 seconds:1
16 44
s
8s
1Area 1 3 4 8
2
The distance is still equal to the area under the curve!
Notice that the area is a trapezoid.
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211
8v t What if:
We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.
It seems reasonable that the distance will equal the area under the curve.
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211
8
dsv t
dt
31
24s t t
314 4
24s
26
3s
The area under the curve2
63
We can use anti-derivatives to find the area under a curve!
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0
1
limn
k kP
k
f c x
b
af x dx
F b F a
Area
“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” -- Gudder
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Area from x=0to x=1
Example: 2y x
Find the area under the curve from x=1 to x=2.
2 2
1x dx
23
1
1
3x
31 12 1
3 3
8 1
3 3
7
3
Area from x=0to x=2
Area under the curve from x=1 to x=2.
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Integrals such as are called definite integrals
because we can find a definite value for the answer.
4 2
1x dx
4 2
1x dx
43
1
1
3x C
3 31 14 1
3 3C C
64 1
3 3C C
63
3 21
The constant always cancels when finding a definite integral, so we leave it out!
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Integrals such as are called indefinite integrals
because we can not find a definite value for the answer.
2x dx
2x dx31
3x C
When finding indefinite integrals, we always include the “plus C”.
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Definite Integration and Areas
0 1
23 2 xy
It can be used to find an area bounded, in part, by a curve
e.g.
1
0
2 23 dxx gives the area shaded on the graph
The limits of integration . . .
Definite integration results in a value.
Areas
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Definite Integration and Areas
. . . give the boundaries of the area.
The limits of integration . . .
0 1
23 2 xy
It can be used to find an area bounded, in part, by a curve
Definite integration results in a value.
Areas
x = 0 is the lower limit( the left hand
boundary )x = 1 is the upper limit(the right hand
boundary )
dxx 23 2
0
1
e.g.
gives the area shaded on the graph
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Definite Integration and Areas
. . . give the boundaries of the area.
The limits of integration . . .
0 1
23 2 xy
It can be used to find an area bounded, in part, by a curve
Definite integration results in a value.
Areas
x = 0 is the lower limit( the left hand
boundary )x = 1 is the upper limit(the right hand
boundary )
dxx 23 2
0
1
e.g.
gives the area shaded on the graph
![Page 25: Summation Notation Also called sigma notationAlso called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series 2+4+6+8+10 can be written.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649ee15503460f94bf2526/html5/thumbnails/25.jpg)
Definite Integration and Areas
0 1
23 2 xy
Finding an area
the shaded area equals 3
The units are usually unknown in this type of question
1
0
2 23 dxxSince
31
0
xx 23
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Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals1)1) Order of Integration:Order of Integration:
( ) ( )a b
b af x dx f x dx
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Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals2)2) Zero:Zero: ( ) 0
a
af x dx
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Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals3) Constant Multiple:3) Constant Multiple:
( ) ( )
( ) ( )
b b
a a
b b
a a
kf x dx k f x dx
f x dx f x dx
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Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals4) Sum and Difference:4) Sum and Difference:
( ( ) ( )) ( ) ( )b b b
a a af x g x dx f x dx g x dx
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Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals
5) Additivity:5) Additivity:
( ) ( ) ( )b c c
a b af x dx f x dx f x dx
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Definite Integration and Areas
Using the rules for Using the rules for integrationintegrationSuppose:Suppose:
Find each of the following integrals, if Find each of the following integrals, if possiblepossible::
a)a) b)b) c) c)
d)d) e) e) f) f)
1
1( ) 5f x dx
4
1( ) 2f x dx
1
1( ) 7h x dx
1
4( )f x dx
4
1( )f x dx
1
12 ( ) 3 ( )f x h x dx
1
0( )f x dx
2
2( )h x dx
4
1( ) ( )f x h x dx
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Definite Integration and Areas
“Thus mathematics may be defined as the subject in which we never know what we are talking about, not whether what we are saying is true.” -- Russell
3
2
2xdx20
12
dx3
2
1
3x dx3
1
(2 3)x dx5
3 2
2
( )x x dx1
0
sin d 1
33
2dr
r
22
1
2 3x dx 2 3
0
23
xx dx