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Summation Notation Summation Notation Also called Also called sigma notation sigma notation (sigma is a Greek letter (sigma is a Greek letter Σ Σ meaning meaning “sum”) “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 summation index of summation Sometimes you will see an n or k here Sometimes you will see an n or k here instead of i. instead 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 2 i i goes from i goes from 1 1 to 5. to 5.

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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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Page 1: Summation Notation

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.

Page 2: Summation Notation

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

Page 3: Summation Notation

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

Page 4: Summation Notation

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

Page 5: Summation Notation

Estimating with Finite Sums

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

Greenfield Village, Michigan

Page 6: Summation Notation

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

Page 7: Summation Notation

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

Page 8: Summation Notation

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

Page 9: Summation Notation

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

Page 10: Summation Notation

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

Page 11: Summation Notation

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

Page 12: Summation Notation

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

Page 13: Summation Notation

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.

Page 14: Summation Notation

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

Page 15: Summation Notation

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.

Page 16: Summation Notation

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.

Page 17: Summation Notation

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!

Page 18: Summation Notation

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

Page 19: Summation Notation

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.

Page 20: Summation Notation

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!

Page 21: Summation Notation

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”.

Page 22: Summation Notation

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

Page 23: Summation Notation

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 24: Summation Notation

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

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

Page 26: Summation Notation

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

Page 27: Summation Notation

Definite Integration and Areas

Rules for Definite IntegralsRules for Definite Integrals2)2) Zero:Zero: ( ) 0

a

af x dx

Page 28: Summation Notation

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

Page 29: Summation Notation

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

Page 30: Summation Notation

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

Page 31: Summation Notation

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

Page 32: Summation Notation

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