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Summation Notation. Also called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series 2+4+6+8+10 can be written as: i is called the index of summation Sometimes you will see an n or k here instead of i. The notation is read: “the sum from i=1 to 5 of 2i”. i goes from 1 - PowerPoint PPT Presentation
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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.
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
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
...54
43
32
21 . b
...14
413
312
211
1
. to1 from goes where1
Or,
iii
1 1ii
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
Estimating with Finite Sums
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
Greenfield Village, Michigan
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 ftsec
3t d
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.)
21 18
V 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).
1 118
112
128t v
101 11
82 11
2
3 128
Approximate area: 1 1 1 31 1 1 2 5 5.758 2 8 4
We could also use a Right-hand Rectangular Approximation Method (RRAM).
118
112
128
Approximate area: 1 1 1 31 1 2 3 7 7.758 2 8 4
3
21 18
V t
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.
21 18
V t
21 18
V 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
When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.
21 18
V 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
1
limn
kn k
f c x
Leibnitz introduced a simpler notation for the definite integral:
1
limn b
k an k
f c x f x dx
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.
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 ftsec
If the velocity varies:
1 12
v 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 82
The distance is still equal to the area under the curve!
Notice that the area is a trapezoid.
21 18
v 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.
21 18
dsv tdt
3124
s t t
31 4 424
s
263
s
The area under the curve263
We can use anti-derivatives to find the area under a curve!
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
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
13x
31 12 13 3
8 13 3
73
Area from x=0to x=2
Area under the curve from x=1 to x=2.
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
13x C
3 31 14 13 3
C C
64 13 3
C C 633
21
The constant always cancels when finding a definite integral, so we leave it out!
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”.
Definite Integration and Areas
0 1
23 2 xy
It can be used to find an area bounded, in part, by a curvee.g.
1
0
2 23 dxx gives the area shaded on the graph
The limits of integration . . .
Definite integration results in a value.
Areas
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
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
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
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
Definite Integration and Areas
Rules for Definite IntegralsRules for Definite Integrals2)2) Zero:Zero: ( ) 0
a
af x dx
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
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
Definite Integration and AreasRules for Definite IntegralsRules for Definite Integrals
5) Additivity:5) Additivity:
( ) ( ) ( )b c c
a b af x dx f x dx f x dx
Definite Integration and AreasUsing 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
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
2 drr
22
1
2 3x dx 2 3
0
23x x dx