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4.2 Area
Sigma Notation
Summation ExamplesExample:
5
1i
i
Example:
3
1
2
j
j
Example:
4
1
3k
1 2 3 4 5 15
1 4 9 14
3 3 33 12
Example:
3
1
52i
i 7 9 11 27
Example 1
12 2
0
kn
n
k
More Summation Examples
112 2
n 12
2 2
n 13
2 2
n 10
2 2
n
12 2 nn
......
i
n
ii xxf
1
)( 11)( xxf 22 )( xxf 33)( xxf
......nn xxf )(
Theorem 4.2 Summation Rules
n
iika
1
n
iiak
1
n
iii ba
1
)(
n
iia
1
n
iib
1
n
i
c1
cn
n
i
i1 2
)1( nn
n
i
i1
2
6
)12)(1( nnn
n
i
i1
3
4
)1( 22 nn
Theorem 4.2 Summation Rules
Example 2 Evaluate the summation
100
1
)92(i
i
2100
1i
i
100
1
9i
2
)101(1002 9100
100,10 900 000,11
Solution
100
1
)92(i
i
Examples
Example 3 Compute
4
1
2)13(i
i
4
1
2 )169(i
ii
4
1
29i
i
4
1
6i
i
4
1
1i
6
)9)(5(49
2
)5(46 )1)(4(
270 60 4 214
4
1
2)13(i
i
Solution
Examples
Example 4 Evaluate the summation for n = 100 and 10000
1
22
1n
k n
k
Solution
1
22
1n
k n
k
1
22
)1(1 n
k
kn
n
k
kn 1
2
1
Note that we change (shift) the upper and lower bound
2
)1(12
nn
n n
n
2
1
For n = 100
200
101
2
1
n
n For n = 10000
20000
10001
2
1
n
n
Examples
Summation and LimitsExample 5 Find the limit for
n
n
i nni
1
2 )1()2( lim
n
n
i ni
n 1
2)2(1
lim
n
n
i nii
nn 12
2
)44(1
lim
n
nnn
n
nn
nn
n 6
)12)(1(1
2
)1(44
1 lim
2
n
n
n
n
n
n 121
6
1)1(24 lim
Continued…
n n
n
n
n
n
n 121
6
1)1(24 lim
n nnn
12
11
6
11124 lim
)2)(1(61(2)(1)4
319
316
Area
wlA hbA 2
12rA
2
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
The total number of inscribed rectangles
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
The total number of circumscribed rectangles
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
Upper and Lower Sums
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. The endpoints of the intervals are
babaxnax
xax
xax
axax
n
)()(
)(2
)(1
)(0
2
1
0
Upper and Lower Sums
Because the function f(x) is continuous, the Extreme Value Theorem guarantees the existence of a minimum and a maximum value of f(x) in each subinterval.
We know that the height of the i-th inscribed rectangle is f(mi) and that of circumscribed rectangle is f(Mi).
)(min)(],[ 1
xfmfii xxx
i
)(max)(],[ 1
xfMfii xxx
i
Upper and Lower Sums
The i-th regional area Ai is bounded by the inscribed and circumscribed rectangles.
We know that the relationship among the Lower Sum, area of the region, and the Upper Sum is
xMfAxmf iii )()(
n
ii
n
ii
n
ii xMfAxmf
111
)()(
)()()()(111
nSxMfAAxmfnsn
ii
n
ii
n
ii
Theorem 4.3 Limits of the Upper and Lower Sums
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
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
Regular Right-Endpoint Formula
RR-EFintervals are regular in length
squaring from right endpt of rect.
Example 6 Find the area under the graph of 5] [1, interval on the 64)( 2 xxxf
1 5
a = 1b = 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
HomeworkPg. 267 1, 7, 11, 15, 21, 31, 33, 41, 23-29 odd, 39, 43
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