Sigma Notation, Upper and Lower Sums

Area

Sigma NotationDefinition – a concise notation for sums.This notation is called sigma notation

because it uses the uppercase Greek letter sigma, written as ∑.

The sum of n terms1, 2 3, , . . . is written asna a a a

1 2 31

i

. . . .

where is the , a is the of the sum,

and the are and 1.

n

ni

a a a a

i index of summation ith term

upper and lower bounds of summation n

Examples of Sigma Notation

5

2

52

2

2 2

22

( 1)( 3)

2 3

(2) 2(2) 3 (3) 2(3) 3

(4) 2(4) 3 5 2(5) 3

3 0 5 12 14

k

k

k k

k k

Examples of Sigma Notation

5

3

1

1 1 1

3 4 520 15 12 47

60 60 60 60

j j

Examples of Sigma Notation

42 3

1

2 32 3 2 3

2 3

( 1) ( 1)

(1 1) (1 1) (2 1) (2 1) 3 1 3 1

4 1 (4 1)

0 8 1 27 4 64 9 125

8 28 68 134

238

i

i i

Summation Formulas

Using Formulas to Evaluate a SumEvaluate the following summation for n =

10, 100, 1000 and 10,000.

2 21 1

21 1

2

2

2

2

2 2

1 11 ( )

11

1 ( 1)1( )

2

1 2

2

1 3 ( 3) 3

2 2 2

n n

i i

n n

i i

ii the index of summation is i

n n

in

n nn

n

n n n

n

n n n n n

n n n

Using Formulas to Evaluate a SumNow we have to substitute 10, 100, 1000,

and 10,000 in for n.n = 10 the answer is 0.65000n = 100 the answer is 0.51500n = 1000 the answer is 0.50150n = 10,000 the answer is 0. 50015

What does the answer appear to approach as the n’s get larger and larger (limit as n approaches infinity)?

AreaFinding the area of a polygon is simple

because any plane figure with edges can be broken into rectangles and triangles.

Finding the area of a circular object or curve is not so easy.

In order to find the area, we break the figure into rectangles. The more rectangles, the more accurate the area will be.

Approximating the Area of a Plane RegionUse five rectangles to find two

approximations of the area of the region lying between the graph of

and the x-axis between the graph of x = 0 and x = 2.

2( ) 5f x x

Steps1. Draw the graph2. Find the width of each rectangle by

taking the larger number and subtracting the smaller number. Then divide by the number of rectangles designated.

3. Now find the height by putting the x values found in number 2 into the equation.

4. Multiply the length times the height (to find the area of each rectangle).

5. Add each of these together to find the total area.

Approximating the Area of a Plane Region

2 0To find the width . We now have to know

5the height of each rectangle. To find this, we need to

find ( ) where is 1, 2, 3, 4, and 5 (the number of

2rectangles). Therefore, we need to find (1

5

x

f xi i

f

) ,

2 2 2 2(2) , (3) , (4) , (5) .

5 5 5 5f f f and f

Approximating the Area of a Plane Region

2 2 4 4 6 6 8 80, , , , , , , , , 2

5 5 5 5 5 5 5 5

The right endpoints are the numbers on the right. (These are not

ordered pairs)

The sum of the areas of the five rectangles is:

Approximating the Area of a Plane Region

Now let’s find the area using the left endpoints. The five left endpoints will involve using the i – 1 rectangle. This answer will be too large because there is lots of area being counted that is not included (look at the graph).

2 25 5 5

1 1 1

5 52

1 1

2 2 2 2 4 25 5

5 5 5 5 25 5

8 8 5(5 1)(10 1)2 10 6.48

125 125 6

i i i

i i

if i i

i

Approximating the Area of a Plane Region

5 5

1 1

225 5

1 1

5 5 52

1 1 1

2( 1) 2 2 2 2

5 5 5 5

4 8 42 2 2 25 5

5 5 25 5

8 16 2421

125 125 125

8 5(5 1)(10 1) 16

125 6 125

i i

i i

i i i

i if f

i ii

i i

5(5 1) 2425

2 125

8.08

Approximating the Area of a Plane RegionThe true area must be somewhere

between these two numbers.The area would be more accurate if we

used more rectangles.Let’s use the program from yesterday to

find the area using 10 rectangles, 100 rectangles, and 1000 rectangles.

What do you think the true area is?

Upper and Lower Sums

An inscribed rectangle lies inside the ith regionA circumscribed rectangle lies outside the ith

regionAn area found using an inscribed rectangle is

smaller than the actual areaAn area found using a circumscribed rectangle

is larger than the actual areaThe sum of the areas of the inscribed

rectangles is called a lower sum.The sum of the areas of the circumscribed

rectangles is called an upper sum.

Example of Finding Upper and Lower SumsFind the upper an lower sums for the

region bounded by the graph of Remember to first draw the graph.Next find the width using the formula

2( ) 2 1 and the between 0 2f x x x x axis x and x

2 0 2b ax

n n n

Example of Finding Lower and Upper Sums

1 1

2n

i=1

Left endpoints Right endpoints

2 20 ( 1) 0

(0 because 0)

2 2 2( ) ( )

2 2 2 2 22 1

i i

n n

ii n

m i M in n

a

Lower Sum

is n f m x f

n n

i i

n n n

Example of Finding Lower Sum

2

21

23 3 3 2 2

1 1 1 1 1 1

3 3 3 2 2

3 2 2

3

4 8 4 4 4 21

8 16 8 8 8 21 1 1

8 ( 1)(2 1) 16 ( 1) 8 8 ( 1) 8 2(1) ( ) ( )

6 2 2

4(2 3 ) 8 8

3

n

i

n n n n n n

i i i i i i

i i i

n n n

i i in n n n n n

n n n n n n nn n

n n n n n n

n n n n

n n

2

3 3 2

8 4 48 2

nn

n n

Example of Finding Lower Sum

3 2 3 2 3

3 3 3 3

3 2

3 2

Find a common denominator and combine terms:

8n 12 4 24 24 24 12 12 24 6

3 3 3 3

14 12 8 14 4 8

3 3 3

n n n n n n n

n n n n

n n n

n n n

Finding an Upper SumUsing right endpoints

11 1

22

3 21 1 1 1

3 2

3 2

3 2 2

3 2

3 2

2 2( ) ( )

2 2 2 8 8 22 1 1

8 2 3 8 ( 1) 2( )

6 2

8 12 4 4 42

3

8 12 4 12

n n

i i

n n n n

i i i i

iS n f M x f

n n

i ii i

n n n n n n

n n n n nn

n n n

n n n n n

n n

n n n

3 2 3

3

3 2

3 2

12 6

3

14 24 4 14 8 4

3 3 3

n n n

n

n n n

n n n

Limit of the Lower and Upper SumsLet f be continuous and nonnegative

on the interval [a, b]. The limits as n —›∞ of both

upper and lower sums exist and are equal to each other. That is,lim ( ) ( )

ns n S n

Definition of the Area of a Region in the PlaneLet f be continuous and nonnegative

on the interval[a, b]. The area of the region bounded

by the graph of f, the x-axis, and the vertical lines x = a and x = b is

11

lim

where

n

i i i in

i

Area f c x x c x

b ax

n