11.5 Area2014

After this lesson, you should be able to:

Use sigma notation to write and evaluate a sum.Understand the concept of area.Approximate the area of a plane region.Find the area of a plane region using limits.

ReviewExample : Evaluate the following limit:

2

1

1lim 1

n

ni

i

n n

7

3

Area

wlA hbA 2

12rA

2

2)( xxf

Area of the region bounded by and the lines x=2 and y=0?

2)( xxf

x

y

Lower ApproximationUsing 4 inscribed rectangles of equal width

Lower approximation =(sum of the rectangles)

4

91

4

10

4

2

4

14

2

14

7

2

2)( xxf

x

y

Using 4 circumscribed rectangles of equal width

Upper approximation =(sum of the rectangles)

4

4

91

4

1

4

2

4

30

2

1

4

15

2

2)( xxf

Upper Approximation

Continued…

4

91

4

10

2

1

4

14

2

1

4

7

L

4

4

91

4

1

2

1

4

30

2

1

4

15

U

L A U

4

7 A 4

15The average of the lower and upper approximations is

2

LU

2

415

47

2

422

4

11

A is approximately 4

11

The procedure we just used can be generalized to the methodology to calculate the area of a plane region. We begin with subdividing the interval [a, b] into n subintervals, each of equal width x = (b – a)/n.

Theorem 4.3 Limits of the Upper and Lower Sums

a b

n

abx

Area =

n

n

i n

abi

n

abaf

1

lim

height x base

In General - Finding Area Using the Limit

Or, xi , the i-th right endpoint

x

y

2

2)( xxf

length = 2 – 0 = 2

xnn

202

n = # of rectangles

A

n

n

i nin

f1

22 lim

n

n

i

inn 1

22

42 lim

n

n

i

inn 1

22

42 lim

Exact Area Using the Limit

in

M i

2

A

n

n

i nin

f1

22 lim

n

n

i

inn 1

22

42 lim

n

n

i

inn 1

22

42 lim

n

nnn

nn 6

)12)(1(42 lim

2

n n

n

n

n 121

3

4 lim

n nn

12

11

3

4 lim

)2)(1(3

4

3

8

Exact Area Using the Limit

Definition of the Area of a Region in the Plane

Regular Right-Endpoint Formula

RR-EFExample 6 Find the area under the graph of 2( ) 4 6 on the interval [1, 5]f x x x

1 5

n

ab

nn

415

in

aba i

n

41

A =

n

n

i nin

f1

441 lim

n

n

i

in

inn 1

2

64

144

14

lim

n

n

i

in

in

inn 1

22

616

4168

14

lim

n

n

i nin

fA1

441 lim

n

n

i

in

inn 1

2

64

144

14

lim

n

n

i

in

in

inn 1

22

616

4168

14

lim

n

n

i

in

inn 1

22

38164

lim

nn

nn

n

nnn

nn3

2

)1(8

6

)12)(1(164 lim

2

Regular Right-Endpoint Formula

nn

nn

n

nnn

nn3

2

)1(8

6

)12)(1(164 lim

2

n n

nnn

n12

)1(16)12)(1(

3

32 lim

2

n n

n

n

n

n

n12

116

121

3

32 lim

12)1(16)2)(1(3

32

n nnn12

1116

12

11

3

32 lim

Continued

3

52

Regular Right-Endpoint Formula

RR-EFExample 7 Find the area bounded by the graph of f(x),

the x-axis, the y-axis, and x = 3.

2( ) 9 on the interval [0, 3]f x x

18

Regular Right-Endpoint Formula

RR-EFExample 8 Find the area bounded by the graph of f(x),

and the x-axis on the given interval

2( ) 3 4 on the interval [1, 4]f x x x

21

2

HomeworkDay 1: Section 11.5 pg. 788 1-5 odd, 15-29 odd

Day 2: Section 11.5 pg. 788 16-30 even

Day 3: Ch. 11 Review pg. 791 3-91 odd

Day 4: Ch. 11 Practice Test

Ch. 11 Test Monday 5/11

HWQ

Find the area between the graph of f(x) and the x-axis on the given interval:

2( ) 2 on the interval [0, 1]f x x