4.2 Area

Sigma Notation

n

iia

1

where i is the index of summation, ai is the ith

term, and the lower and upper bounds of summation are 1 and n respectively

The sum of n terms a1, a2, a3, ….an is written

6

1i

i

5

0

)1(i

i

7

3

2

j

j

Examples:

n

ii

n

ii akka

11

n

ii

n

ii

n

iii baba

111

)(

Properties of Summations

cncn

i

1 2

)1(

1

nni

n

i

6

12)1(

1

2

nnni

n

i 4

)1( 22

1

3

nni

n

i

Summation Formulas

Example 1: Find the sum of the first 100 integers.

Example 2: Summation Practice

5

2

31i

ii

30

1

2i

i

20

1

21j

j

Example 3: Limits Review

nnnnn

233

323

4lim

2

118lim

2

nn

nn

Example 4: Limit of a Sequence

n

in n

i

12

16lim

n

in

in1

2

31

1lim

n

in

in1

2

21

1lim

Warm-up

Definition of the Area of a Rectangle: A=bh

Take a rectangle whose area is twice the triangle: A=1/2 bh

For any polygon, just divide the polygon into triangles.

Area of Inscribed Polygon < Area Circle < Area of Circumscribed Polygon

Area of a Plane Region

Find the area under the curve of

5)( 2 xxf

x

y

Between x = 0 and x = 2

Area of a Plane Region—Upper and Lower Sums

n

abx

bxnaxaxaxa ...210

intervalith on the maximum)(

intervalith on the minimum)(

i

i

Mf

mf

Begin by subdividing the interval [a,b] into n subintervals, each of length

Endpoints of the subintervals:

Because f is continuous, the Extreme Value Theorem guarantees the existence of a min and a max on the interval.

Rectangle Rectangle

bedCircumscri of Area)()( Inscribed of Area xMfxmf ii

Rectangle Rectangle

bedCircumscri of Area)()( Inscribed of Area xMfxmf ii

Sum of these areas=

lower sum

Sum of these areas=

upper sum

n

ii

n

ii

xMfnS

xmfns

1

1

)(SumUpper

)(SumLower

x

y

x

y

Example: Find the upper and lower sums for the region bounded by the graph

of 2 and 0between axis theand )( 2 xxxxxf

other.each toequal are sumsupper andlower both the of

as limits The b].[a, interval on the enonnegativ and continuous be fLet

Sums Upper andLower theofLimit :Theorem

n

n

abx

xcxxcf

bxax

iii

n

ii

n

where

,)(limArea

is and lines vertical theand axis,- x thef, ofgraph by the bounded

region theof area The b].[a, interval on the enonnegativ and continuous be fLet

Plane in theRegion of Area theof Definition

11

Example: Find the area of the region bounded by the graph

of 1 and 0 lines vertical theand )( 3 xxxxf

Example: Find the area of the region bounded by the graph

of 3 and 1 lines vertical theand 9)( 2 xxxxf

Example: Find the area of the region bounded by the graph

of 1 and 0 lines horizontal theand )( 2 yyyyf

x

y