area use sigma notation to write and evaluate a sumuse sigma notation to write and evaluate a sum...
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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
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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.
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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.
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Examples (cont.)6
2
2
2
1
1
4.
15. ( 1)
6. ( )
k
n
i
n
ii
k
in
f x x
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VariablesVariables
Although any variable can be used as the index of summation, i, j,i, j, and kk are most often used.
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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
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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
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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
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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
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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!
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Back to our favorite function…
y = xy = x22
What’s the area under this curve from x = 0 to x = 2?
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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
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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
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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
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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)
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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?
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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
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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
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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
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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
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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
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Uh Oh...
• Now what happens if the function doesn’t stay strictly increasing or decreasing and I want to find lower or upper sums?
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Right....there’s LIMITS!
The World is HAPPY!
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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