4.2 Area Under a Curve

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4.2 Area Under a Curve

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Sigma (summation) notation REVIEW5

11 2 3 4 5

kk

In this case k is the index of summation

The lower and upper bounds of summation are 1 and 5

63 3 3 3 3 3 3

11 2 3 4 5 6

ii

In this case i is the index of summation

The lower and upper bounds of summation are 1 and 6

63

11 8 27 64 125 216

ii

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Sigma notation4

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1 2 3 4 1631 2 3 4 5 60k

kk

3

1 1 2 2 3 31

( ) ( ) ( ) ( )k kk

f x x f x x f x x f x x

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Sigma Summation Notation

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Practice with Summation Notation

= 3080

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Practice with Summation NotationNumerical Problems can be done with the TI83+/84 as was done in PreCalc Algebra

Sum is in LIST, MATH

Seq is on LIST, OPS

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TI-89 Calculator Steps• Hit F3, go down to #4 sum• Enter in the equation• Comma x• Comma lower bound• Comma upper bound• End the parenthesisTry: Answer: 10,950

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Area Under a Curve by Limit DefinitionThe area under a curve can be approximated by the sum of rectangles. The figure on the left shows inscribed rectangles while the figure on the right shows circumscribed rectangles

This gives the upper sum.

This gives the lower sum.

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This gives the lower sum.

If the width of each of n rectangles is x, and the height is the minimum value of f in the rectangle, f(Mi), then the area is the limit of the area of the rectangles as n

Area Under a Curve by Limit Definition

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Area under a curve by limit definition

This gives the upper sum.

If the width of each of n rectangles is x, and the height is the maximum value of f in the rectangle, f(mi), then the area is the limit of the area of the rectangles as n

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Area under a curve by limit definition

The limit as n of the Upper Sum =

The limit as n of the Lower Sum =

The area under the curve between x = a and x = b.

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Area under a curve by limit definition

Area =1

lim ( )

n

in i

f c x

b awhere xn

f is continuous on [a,b]

ci is any point in the interval

a b ci

f(ci)

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Visualization

ith interval

f(ci)

Width = Δx

ci

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Example: Area under a curve by limit definition

Find the area of the region bounded by the graph f(x) = 2x – x3 , the x-axis, and the vertical lines x = 0 and x = 1, as shown in the figure.

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Area under a curve by limit definition

Why is right, endpoint i/n?

Suppose the interval from 0 to 1 is divided into 10 subintervals, the endpoint of the first one is 1/10, endpoint of the second one is 2/10 … so the right endpoint of the ith is i/10.

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Visualization again

ith interval

f(ci)

Width = Δx=

ci = i/n

b an

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1

3

1

lim ( )

1lim 2

n

ini

n

ni

Area f c x

i in n n

32 4

1 1

2 1lim

n n

n i i

i in n

Find the area of the region bounded by the graph f(x) = 2x – x3 on [0, 1]

2 2

2 4

2 ( 1) 1 ( 1)lim2 4n

n n n nn n

Sum of all the rectangles

Right endpoint

Sub for x in f(x)

Use rules of summation

ii nc

ii nc

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…continued1

3

1

lim ( )

1lim 2

n

ini

n

ni

Area f c x

i in n n

32 4

1 1

2 2limn n

ni i

i in n

2

2

( 1) ( 1)lim4n

n nn n

2 2

2 4

2 ( 1) 1 ( 1)lim2 4n

n n n nn n

2

1 1 1 1lim 14 2 4n n n n

1 314 4

Foil & Simplify

2

2

1 2 1lim 14n

n nn n

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The area of the region bounded by the graph f(x) = 2x – x3 , the x-axis, and the vertical lines x = 0 and x = 1, as shown in the figure = .75

0.75

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Practice with Limits

2

2

1 ( )lim2n

n nn

1 1lim 12n n

2

1 ( 1)lim2n

n nn

1 11 02 2

Multiply out

Separate

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4.2 Area

Please read the text and go over the examples in the

text. Carefully do the assignment making sure you can work out the summation notation. This is an important section though we only have

one day to spend on it.Assignment: p 267-269 #1-65

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