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

Section 5.2

SIGMA NotationSIGMA NotationThe sum of n terms a1 , a2 , a3 ,…, an

is written as

1 2 3 41

...n

i ni

a a a a a a

Where ii is the index of summationindex of summation, aaii is the ithith termterm of the sum, and the upperupper and lowerlower bounds of summation are nn and 11.

Note: ii does not have to be 1. Any integer less than or equal to the upper bound is legitimate.

Examples6

0

7

1

4

2

1.

2. ( 1)

3. ( 2)

i

j

k

i

j

k

Note:Note: The same sum can be represented in different ways using sigma notation.

Examples (cont.)6

2

2

2

1

1

4.

15. ( 1)

6. ( )

k

n

i

n

ii

k

in

f x x

VariablesVariables

Although any variable can be used as the index of summation, i, j,i, j, and kk are most often used.

Summation PropertiesSummation Properties

1 1

when k is a constantn n

i ii i

ka k a

1 1 1

( ) n n n

i i i ii i i

a b a b

Summation Formulas

2

1

1

1

1

3

1.

2.

3.

4.

n

i

n

i

n

i

n

i

c

i

i

i

cn

( 1)

2

n n

Theorem

( 1)(2 1)

6

n n n

2 2( 1)

4

n n

Try this one….Try this one…. 21

1n

i

i

n

21

1n

i

i

n

21

1( 1)

n

i

in

Evaluate:

21 1

11

n n

i i

in

2

1 ( 1)

2

n nn

n

2

2

1 3

2

n n

n

3

2

n

n

For n = 10, 100, 1000, 10,000

So, ….So, ….2

1

1n

i

i

n

21

1n

i

i

n

n

10

100

1000

10,000

Evaluate:

3

2

n

n

For n = 10, 100, 1000, 10,000

21

1n

i

i

n

0.65000

0.51500

0.501500.50015What do you think this limit is?

3lim

2x

n

n

.5

Area

Time out for a

construction!

Now on to

What’s summation got to do with it?

This construction is brought to you by the Greek Mathematician Archimedes!

Back to our favorite function…

y = xy = x22

What’s the area under this curve from x = 0 to x = 2?

y = xy = x22 What’s the area under the curve from x = 0 to x = 2? Let’s approximate it with rectangles…

n = 2 x = _____

Inscribed rectangles

Ht. of rect. 1?________

Ht. of rect. 2? ________

Lower sum =

x = B – A n

4

3

2

1

1 2

(x,f(B))

(x,f(A))1A B

y = xy = x22 What’s the area under the curve from x = 0 to x = 2? Let’s approximate it with rectangles…

n = 2 x = _____

Circumscribed rectangles

Ht. of rect. 1?________

Ht. of rect. 2? ________

Upper sum =

x = B – A n

4

3

2

1

1 2

(x,f(B))

(x,f(A))1A B

y = xy = x22 What’s the area under the curve from x = 0 to x = 2? Let’s approximate it with rectangles…

n = 4 x = _____Inscribed rectanglesLower sum =

n = 4 x = _____ Circumscribed rectanglesUpper sum =

x= B – A n

5

4

3

2

1

1 2

(x,f(B))

(x,f(A))1A B

5

4

3

2

1

1 2

(x,f(B))

(x,f(A))1A B

So, what is the area under y = x2 from x = 0 to x = 2?• Read and take notes on section 5.2

(p. 295 – 303)

• Use sigma notation and limits to find this area after reading 5.2! Be able to explain and discuss this tomorrow!

• Do p. 303 # multiple of 3’s from 3 to 45 (i.e. 3, 6, 9, 12, …, 39, 42, 45)

Limits, Sigmas and all Limits, Sigmas and all that Jazz….that Jazz….

1. We can find the area under a curve with the help of sigma notation and limits. We can also approximate area with rectangles.

2. Lower sums are found by summing up the areas of inscribed rectangles where n is the number of rectangles, x is the width of each rectangle, and f(mi) is the height of each inscribed rectangle.

3. Upper sums are found by summing up the areas of circumscribed rectangles where n is the number of rectangles, x is the width of each rectangle, and f(Mi) is the height of each circumscribed rectangle.

4. As the number of rectangles approach infinity, the lower sums = the upper sums.

In other words…..What have we learned from Archimedes?

Left vs. Right EndpointsLeft vs. Right Endpoints

• Find ∆x which is the width of each partition on the interval [A, B]

• The left endpoints can be found using the following formula if i = 1. Why? A + (i -1) x

• The right endpoints can be found using the following formula if i = 1. Why? A + (i) x

x= B – A n

Increasing Functions:Increasing Functions: Upper vs. lower sumsUpper vs. lower sums

• Lower Sum – Inscribed Rectangles

3

2.5

2

1.5

1

0.5

1

(x,f(B))

(x,f(A))

1A B

In each rectangle which endpoint is used in the function to determine the height of the rectangle?

1

1

1

rectan rectanglegle

( 1)

( 1)

n

i

n

i

n

i

height of

B Af A i

n

f A i x

width of

x

x

Increasing Functions:Increasing Functions: Upper vs. Lower sumsUpper vs. Lower sums

• Upper Sum – Circumscribed Rectangles 3

2.5

2

1.5

1

0.5

1 2

(x,f(B))

(x,f(A))

1A B

In each rectangle which endpoint is used in the function to determine the height of the rectangle?

1

1

1

rectan rectanglg

(

ele

)

( )

n

i

n

i

n

i

height of

B

width o

Af A i

n

f A i x

f

x

x

Is this true for Decreasing functions also?

1 2

(x,f(B))

(x,f(A))

1A B

• Lower Sum – Inscribed Rectangles

In each rectangle which endpoint is used in the function to determine the height of the rectangle?

1

1

1

rectan rectanglg

(

ele

)

( )

n

i

n

i

n

i

height of

B

width o

Af A i

n

f A i x

f

x

x

Decreasing Functions:Decreasing Functions: Upper Upper vs. Lower sumsvs. Lower sums

• Upper Sum – Circumscribed Rectangles

1 2

(x,f(B))

(x,f(A))

1A B

In each rectangle which endpoint is used in the function to determine the height of the rectangle?

1

1

1

rectan rectanglegle

( 1)

( 1)

n

i

n

i

n

i

height of

B Af A i

n

f A i x

width of

x

x

Uh Oh...

• Now what happens if the function doesn’t stay strictly increasing or decreasing and I want to find lower or upper sums?

Right....there’s LIMITS!

The World is HAPPY!

Limits to the Rescue!Who would have guessed?

1

1

li (

m )

n

in

i

i i i

Area

x c x

f c x

where ∆x = (B-A)/n

Homework: (yep that’s right....time to practice)

P. 304 # 46, 49, 52, ...., 73, 74 - 77